Putting Mathematics to Work
 Elementary School The student conducts at least one large scale project each year, beginning in fourth grade, drawn from the following kinds and, over the course of elementary school, conducts projects drawn from at least two of the kinds. A single project may draw on more than one kind. a Data study, in which the student: • develops a question and a hypothesis in a situation where data could help make a decision or recommendation; • decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers; • collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions; • writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers. b Science study, in which the student: • decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement; • develops a prediction (a hypothesis) and develops procedures to test the hypothesis; • collects and records data, represents and displays data, and compares results with predictions; • writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers. c Design of a physical structure, in which the student: • decides on a structure to design, the size and budget constraints, and the scale of design; • makes a first draft of the design, and revises and improves the design in response to input from peers and teachers; • makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers. d Management and planning, in which the student: • decides on what to manage or plan, and the criteria to be used to see if the plan worked; • identifies unexpected events that could disrupt the plan and further plans for such contingencies; • identifies resources needed, e.g., materials, money, time, space, and other people; • writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers; • carries out the plan (optional); • writes a report on the plan that includes resources, budget, and schedule, and acknowledges assistance received from parents, peers, and teachers. • writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers. e Pure mathematics investigation, in which the student: • decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns; • describes a question or concept to investigate; • decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models; • carries out the investigation; • writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.

 Middle School The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds. A single investigation or project may draw on more than one kind. a Data study based on civic, economic, or social issues, in which the student: • selects an issue to investigate; • makes a hypothesis on an expected finding, if appropriate; • gathers data; • analyzes the data using concepts from Standard 4, e.g., considering mean and median, and the frequency and distribution of the data; • shows how the study’s results compare with the hypothesis; • uses pertinent statistics to summarize; • prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings. b Mathematical model of physical phenomena, often used in science studies, in which the student: • carries out a study of a physical system using a mathematical representation of the structure; • uses understanding from Standard 3, particularly with respect to the determination of the function governing behavior in the model; • generalizes about the structure with a rule, i.e., a function, that clearly applies to the phenomenon and goes beyond statistical analysis of a pattern of numbers generated by the situation; • prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings. c Design of a physical structure, in which the student: • generates a plan to build something of value, not necessarily monetary value; • uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes; • summarizes the important features of the structure; • prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings. d Management and planning, in which the student: • determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling; • notes any constraints that will affect the plan; • determines a plan; • uses concepts from any of Standards 1 to 4, depending on the nature of the project; • considers the possibility of a more efficient solution; • prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan. e Pure mathematics investigation, in which the student: • extends or “plays with,” as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers; • uses concepts from any of Standards 1 to 4, e.g., an investigation of Pascal’s triangle would have roots in Standard 1 but could tie in concepts from geometry, algebra, and probability; investigations of derivations of geometric formulas would be rooted in Standard 2 but could require algebra; • determines and expresses generalizations from patterns; • makes conjectures on apparent properties and argues, short of formal proof, why they seem true; • prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

 High School The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of high school, conducts investigations or projects drawn from at least three of the kinds. A single investigation or project may draw on more than one kind. a Data study, in which the student: • carries out a study of data relevant to current civic, economic, scientific, health, or social issues; • uses methods of statistical inference to generalize from the data; • prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information. b Mathematical model of a physical system or phenomenon, in which the student: • carries out a study of a physical system or phenomenon by constructing a mathematical model based on functions to make generalizations about the structure of the system; • uses structural analysis (a direct analysis of the structure of the system) rather than numerical or statistical analysis (an analysis of data about the system); • prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information. c Design of a physical structure, in which the student: • creates a design for a physical structure; • uses general mathematical ideas and techniques to discuss specifications for building the structure; • prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information. d Management and planning analysis, in which the student: • carries out a study of a business or public policy situation involving issues such as optimization, cost-benefit projections, and risks; • uses decision rules and strategies both to analyze options and balance trade-offs; and brings in mathematical ideas that serve to generalize the analysis across different conditions; • prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information. e Pure mathematics investigation, in which the student: • carries out a mathematical investigation of a phenomenon or concept in pure mathematics; • uses methods of mathematical reasoning and justification to make generalizations about the phenomenon; • prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information. f History of a mathematical idea, in which the student: • carries out a historical study tracing the development of a mathematical concept and the people who contributed to it; • includes a discussion of the actual mathematical content and its place in the curriculum of the present day; • prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.