$compare using benchmark fractions$

$compare using benchmark fractions$

They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Mathematically proficient students look closely to discern a pattern or structure.

$compare using benchmark fractions$

They are able to use technological tools to explore and deepen their understanding of concepts. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They detect possible errors by strategically using estimation and other mathematical knowledge.

$compare using benchmark fractions$

For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. 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.

$compare using benchmark fractions$

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.

$compare using benchmark fractions$

Compare using benchmark fractions