# Model with Mathematics

1. identify a problem,
2. study the scenario that gave rise to the problem to determine what the important factors or variables are, interpret these mathematically, identify variables in the situation and selecting those that represent essential features,
3. observe the nature of the data—looking for regularities (e.g., if the data is increasing at a constant rate, it may be modeled by a linear system; if the change is constant at the second level of iteration, it can be modeled by a quadratic function; if each entry in the data is a constant multiple of the previous entry, it is modeled by a geometric/exponential function, etc.),
4. develop and formulate a tentative mathematical model by selecting arithmetical, geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
5. use the model to analyze the problem situation mathematically, draw conclusions, and assess them for reasonableness of the solution,
6. analyze and perform operations on these relationships obtained by the modeling process to draw conclusions,
7. test the solution to determine whether it makes sense in the context of the problem situation,
8. interpret the results of the mathematics in terms of the original situation, and
9. if the solution makes sense and they have a mathematical model for this type of problem, validate the conclusions by comparing them with the situation, then either improve the model or express the model formally in mathematical terms – if it is acceptable, and report the conclusions and the reasoning behind them.
• simplify a complex problem and identify important quantities to look at relationships and they can represent this problem mathematically,
• ask what mathematics do I know to describe this situation either with an equation or a diagram and interpret the results of a mathematical situation,
• look for the mathematics learned to apply to another problem and try to solve the problem by changing the parameters of in the problem.
• What model (quantitative, geometrical, algebraic, statistical, probabilistic, or mixed) could be constructed to represent the problem?
• What are the ways to represent the information in the problem (e.g., create a diagram, graph, table, equations, etc.)?
• What tools and approaches are appropriate to the problem at hand?
• How to select and decide which argument makes sense and is reasonable in the context of the problem?
• How to justify the appropriateness of the solution, explain why it makes sense, and how to convince the group of the reasonableness of the solution?
• How to make sure that the results make sense?
• How to improve/revise the model?
• How do I incorporate the comments and concerns of others in the approach?
• What is the best way of presenting the solution to others?
• What further extensions, generalizations, investigations might be interesting or necessary?
In order to support such learners, the classroom must be a place that encourages choice and provides positive feedback regarding competence.  Teachers in such classrooms:
• Assure all students that they are capable and competent, and their ideas are worth sharing with others and encourage student collaboration,
• Presents problems that encourage student initiative and provide the opportunity for a variety of approaches and representation,
• Make available appropriate manipulatives and instructional materials for exploration,
• Practice and integrates the three roles: didactic, Socratic, and coaching,
• Spend less time talking and more time listening to student questions and reasoning,
• Ask more questions, give measured and focused feedback without curtailing creativity and initiatives, seek suggestions for improving solutions, encourage alternative solution approaches.
• Does this situation represent a linear system? Why do you think so? If so, represent it as a linear system. If not, why not?
• Use the information above to write two ordered pairs (x, y), where x represents the time (in hours) since the cake was removed from the oven and y represents the temperature (in degrees Fahrenheit) of the cake at that time.
• If it is a linear system, write the linear relationship between x and y, in any of the following forms, with general values (two point form; a point and slope form; slope-intercept form; standard form)
• Find the slope of the line through the two points identified in step 2.
• Write the linear equation in slope-intercept form or point-slope form.
• Use the equation from step 5 to estimate the temperature of the cake after 1 hour, after 2-hours, and after 4 hours.
The problem, as given in the book, was straightforward; however, the questions above are reformulated to make sure that the students not only understand the problem but also have a deeper understanding and make connections between different concepts and relate the problem to a realistic situation. After this, the teacher made this problem even more rigorous by asking a series of further questions:
• Why do you think the information given to you in the problem satisfies the conditions of a linear relationship?
• Under what conditions can this be modeled by a linear relationship?
• You know from geometry that two points determine a line, is that condition satisfied here?
• What does a linear relationship look like in general?
• What minimum conditions do you need to be able to find the linear relationship in this situation?
• What is unknown in the linear function you just gave?
• What is unknown in this equation?
• When you look at your ordered pairs, will the slope be positive or negative?
• What will be the orientation of the line?
• How will you find the slope of this line?
• What is the formula for slope?
• Can you find the slope geometrically?
• You said: “The formula for slope is .”
• What do m, (y2−y1) and (x2−x1) represent in the formula?
• What do y2, y1, x2, x1 represent?
• Will the formula…give the same slope for your line?
• Why do you think so?
• Can you prove that the two formulas represent the same slope?
• Please draw a rough sketch of the line.
• Based on this sketch, what can you predict about the temperature in the future?
Then the teacher asked his students to solve the problem. Students calculated the slope by considering two points (two ordered pairs): (0, 370) and (3, 70). As the teacher was walking around in the room looking at their work, he asked students:
• What does -100 mean here?
• What will be the temperature in five hours? 10 hours?
At this point there was a great deal of discussion amongst students and they began to question whether it was really a linear model. Students came to the conclusion that it was a linear model only till the temperature of the cake reached room temperature, and after that it was not a linear model. The teacher introduced several examples of non-linear and mixed models. Students even brought the idea of a step function. This is an example of teaching with rigor, making connections, and how mathematics is used to model real-world prolems. The teacher focused only on one problem during the lesson, but students understood the concept at a deeper level rather than solving several problems just applying a procedure. The requirements of rigor—understanding, fluency, and ability to apply, are parallel to our expectations in reading. A child is a good reader when he or she (a) has acquired fluency in reading (displays speed in decoding, chunking, blending of sounds using efficient strategies indicating phonemic awareness, and word attack), (b) shows comprehension (understands the context, intent, and nuances of meaning in the material read), and (c) is able to use it in real life with confidence (pragmatics—able to read a diversity of materials from different genres and reads for interest and purpose). Mastery in any of these elements alone is not enough because reading is the integration of these skills. Similarly, rigor in mathematics means a student demonstrates intra- and inter-conceptual understanding, fluency in performing computational procedures and their interrelationships, knowledge of the appropriateness of a particular mathematical conceptual and procedural tool, and ability to apply mathematics concepts and procedures in solving meaningful, real-life problems. Finally, it is demonstrated in their ability to communicate this understanding. To achieve the same level of mastery as in reading, mathematics educators need to balance these elements in expectations, instruction, and assessments. The writers of the CCSS-M were careful to balance conceptual understanding, procedural skill and fluency, and application at each grade level.