Standard 4: Model with mathematics

The Standard:
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.

Classroom Observations:
Teachers who are developing students’ capacity to "model with mathematics" move explicitly between real-world scenarios and mathematical representations of those scenarios. A middle childhood teacher might pose a scenario of candy boxes containing multiple flavors to help students identify proportions and ratios of flavors and ingredients. An early adolescence teacher might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted. A teacher of adolescents and young adults might pose a "kite factory" scenario, in which advanced students are asked to determine the conditions for always creating a particular shape of kite given the dimensions of the diagonals and the angle of intersection. Visit the video excerpts below to view multiple examples of teachers engaging students in mathematical modeling.

## Connections to Classroom Practices

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... 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.

Liz O'Neill works with her first grade students engaging them in composing and decomposing numbers within twenty. Her students play a game called "How Many are Hiding?" Pairs are given a bag with 10 cubes, a paper plate, and the "How Many Are Hiding Recording Sheet." One partner takes some of the cubes and "hides" them under the plate. The remaining are placed on the top. The second partner uses sentence frames to answer the questions "What number do you see?", "How many are hiding?", "How do you know __ are hiding"? In addition, the answers are recorded. Roles are then reversed. The partner game gives students practice in composing and decomposing numbers within ten. Students who have completed the original task are challenged with changing the total number of cubes to 15 or 20.

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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... They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense...

Hillary Lewis-Wolfsen invites students to examine a problem about proportions and ratios with a strategy used by a student to organize the information in the problem. In this clip, she gives students “private think time” to work the problem again, refreshing their memories about the problem, asks them to use their “think sheets” to record their ideas, then has them turn to a partner and share and defend their thinking. In the discussion, the pairs share where they think the example student’s strategy reflected misunderstandings of the quantities.

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See this video in the context of an entire lesson.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as ...tables.... They can analyze those relationships mathematically to draw conclusions.

Cecilio Dimas leads a lesson on making comparisons between three different financial plans, helping students use multiple representations of mathematical problems: verbal, tabular, graphical, and algebraic generalization. In this clip, Dimas connects to the prior day’s lesson, in which the class “started a conversation about the economic status of our world [and] about making responsible decisions when we’re spending our money.” His students share that they were to represent various DVD rental plans using verbal and tabular representations.

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See this video in the context of an entire lesson.