4. Model with mathematics

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.

Representative IMP Year 1 Lesson:

Ox Expressions and Ox Expressions at Home (Overland Trail), 36, 38

Students are given a list of variables and their definition. Students are asked to model real life situations with the variables such as “the amount of water consumed in a day by a family” or “the number of people in a wagon train.” The students are also asked to make sense of expressions written with variables. In some cases, the expressions that they are given are nonsense and students must explain why putting two variables together doesn’t necessary make for a sensible expression.