![]() ![]() These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Mathematically proficient students consider the available tools when solving a mathematical problem. 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. They can analyze those relationships mathematically to draw conclusions. 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. 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. 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. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. ![]() In early grades, this might be as simple as writing an addition equation to describe a situation. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Later, students learn to determine domains to which an argument applies. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. They justify their conclusions, communicate them to others, and respond to the arguments of others. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. ![]() They make conjectures and build a logical progression of statements to explore the truth of their conjectures. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. MI.3 Mathematical practicesĬonstruct viable arguments and critique the reasoning of others. ![]()
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