Organizing Principles About Mathematical Modeling

The purpose of mathematical modeling in school mathematics courses is for students to understand that they can use math to better understand things they are interested in in the world.

Mathematical modeling is different from solving word problems. It often feels like initially you are not given enough information to answer the question. There should be room to interpret the problem. There ought to be a range of acceptable assumptions and answers. Modeling requires genuine choices to be made by the modeler.

It is expected that students have support from their teacher and classmates while modeling with mathematics. It is not a solitary activity. Assessment should focus on feedback that helps students improve their modeling skills.
Things the Modeler Does When Modeling with Mathematics
Pose a problem that can be explored with quantitative methods. Identify variables in the situation and select those that represent essential features.
Formulate a model: create and select geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between variables
Compute: Analyze these relationships and perform computations to draw conclusions
Interpret the conclusions in terms of the original situation
Validate the conclusions by comparing them with the situation. Iterate if necessary to improve the model
Report the conclusions and the reasoning behind them
Which Math Classes Will Colleges Expect You To Have Taken
Like high schools, most colleges require applicants to have completed three years of math and recommend four years. Selective colleges often require four years of math, and some schools may also require the completion of particular math classes like algebra 2, geometry, or precalculus.
For students planning on majoring in humanities, the social sciences, or a similar field, the math classes you took in high school will not be as important to colleges because they’ll be looking more at the classes that relate to your intended major. This means you don’t have to take the most challenging math classes your high school offers, although colleges want fundamentally talented and wellrounded people, so you should still aim to do well in the math classes you do take in order to maintain a solid GPA.
If you plan on majoring in a STEM field , expectations will be higher because math skills will be more critical to your college courses and future career. Most colleges will require you to have taken four years of math in high school, sometimes including precalculus and calculus. You’ll be competing for college offers with many other smart STEM people, so you’ll want to help yourself stand out by taking rigorous math classes that are offered at a high level.
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Everyone should take four years of math in high school
How To Prepare And Conduct The Modeling Lesson Or Project

Have data ready to share if you plan to give it when students ask.

Ensure students have access to tools they might be expected to use.

If desired, instruct students to use a template for organizing modeling work.

Whether doing the prompt as a classroom lesson or giving as a project, plan to do the inclass launch in class.
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Ideas For Setting Up An Environment Conducive To Modeling

Provide plenty of blank whiteboard or chalkboard space for groups to work together comfortably. Vertical nonpermanent surfaces are most conducive to productive collaborative work. Vertical means on a vertical wall is better than horizontally on a tabletop, and nonpermanent means something like a dry erase board is better than something like chart paper .

Ensure that students have easy access to any tools that might be useful for the task. These might include:

A supply table containing geometry tools, calculators, scratch paper, graph paper, dry erase markers, postits

Electronic devices to access digital tools
Think about how you will help students manage the time that is available to work on the task. For example:
For lessons, display a countdown timer for intermittent points in the class when you will ask each group to summarize their progress
For lessons, decide what time you will ask groups to transition to writing down their findings in a somewhat organized way
For projects, set some intermediate milestone deadlines to help students know if they are on track.
Why Should My School Or District Choose Modeling Our World With Mathematics
For students who intend to go on to posthigh school learning that requires mathematics MOWWM is recommended as a supportive transition to support their High School and Beyond Plan . The mathematics that students complete in MOWWM will strengthen their skill base to make additional study of mathematics more attainable. Therefore, juniors who take MOWWM may find greater success than would have been expected if they had gone straight from Algebra I and Geometry.
Students success in Modeling Our World with Mathematics may encourage students to enroll in a 4th credit of math, increasing their mathematics facility and expanding their career pathways.
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Levels Of Math Classes During Middle School
 Grade 6 = Here students will be thought about Algebra, Geometry, expressions, relationships, variables, and proportionality.
 Grade 7 = Aside from Algebra and Geometry, students can also learn about inequalities as well as the computation of volume and surface areas of different shapes.
 Grade 8 = Again, Algebra and Geometry are thought here with the addition of linear functions, graphing, and others.
High School Trigonometry Curriculum
Time4Learnings onesemester trigonometry course helps students use their geometry and algebra skills to begin their study of trigonometry. Students will be required to express understanding using qualitative, quantitative, algebraic, and graphing skills. This course begins with a quick overview of right triangle relationships before introducing trigonometric functions and their applications. Students explore angles and radian measures, circular trigonometry, and the unit circle. Students extend their understanding to trigonometric graphs, including the effects of translations and the inverses of trigonometric functions. Onscreen teachers explain the mathematical theory of each concept and detail the stepby step procedures for each algorithm.
Learn more about the high school trigonometry curriculum.
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These Maths Projects Help In Developing Very Important Mathematical Skills Like:

Correlating the concepts taught in the classes with the practical applications of those concepts

Proving a hand on experience to the children

Fostering teamwork, coordination, and communication along with creativity and knowledge

Understanding and grasping the ideas of mathematics in a better way

Visualizing the concepts in the form of diagrams, graphs, and images facilitates a better understanding

Improving their problemsolving skills, reasoning, and planning skills, etc.

Making reallife decisions that leads to a holistic approach to learning.
In this article, we have brought for you 20 topics for maths projects which will help you develop simple maths projects.
The High School Math Courses You Should Take
Choosing which math classes to study can be one of the most challenging parts of planning your high school schedule. High schools offer numerous math classes, often at varying degrees of difficulty, and it can be difficult to know which math classes will be the best for you and your future.
Read this guide to learn about standard high school math curriculum, AP and IB math courses, which math classes colleges expect you to have taken, and ways to exceed those expectations.
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Math Models With Applications
Math Models is an appllications course with financial emphasis. Students will investigate decision making related to earning, spending, borrowing, and investing money. MMA reinforces skills learned in Algebra 1 and Geometry. It prepares students to be more successful in Algebra 2.
Prerequisites: Algebra 1 & Geometry
Ap Computer Science A
This course serves as an introduction to computers and the study of managing and processing information. The emphasis is on solving real world problems by means of computer programming . Students will learn thoroughly the Java programming language and apply those skills in exploring how computers work. Some topics covered include objectoriented techniques, file management, data structures, classes, objects, graphics, debugging, hardware components, and social implications. The course includes an in depth treatment of the AP Simulation Case Study. Students will have the option of taking the AP exam for which many colleges will grant up to 3 hours of college credit. Successful completion of both semesters of this course will satisfy a mathematics credit.
Prerequisite: Algebra II or concurrent enrollment in Algebra II
What’s next? Computer Science III Honors
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High School Algebra I Curriculum
Typically taken in 9th grade, Time4Learnings high school Algebra I course is a comprehensive introduction to the relationships between quantities and reasoning with equations, linear and exponential relationships, descriptive statistics, expressions and equations, and quadratic functions and modeling. This course builds on the foundation set in middle grades by deepening students understanding of linear and exponential functions and developing fluency in writing and solving onevariable equations and inequalities. With tools like a digital notebook for taking notes, embedded calculators to help work through difficult problems, and the ability to watch video instruction as many times as needed, students are empowered to take ownership of their Algebra I learning.
Learn more about our high school Algebra I curriculum.
Theory Of Change Underlying The Model
The hybrid course model seeks to overcome three obstacles to learning:
 Weak student prerequisite knowledge upon course entry
 Passivity in the students’ instructional experience
 Inconsistent and poorquality instructional practice in large multisection classes
Our interactive video presentation describes how these three problems are directly addressed through the five model components:
As Figure 4 shows, the structure and relationship of the five components facilitates the “learning flow” that moves students actively and successfully from classroom work, to group work, to homework, to exams.
Figure 4. How Students Flow through Math 103/103L
Once in place, the flow created by the five components and the instructional staff’s coordinated efforts gradually moves the student from passive listener to independent learner. In the hybrid model, students constantly refresh remedial knowledge/skills so as to:
 Arrive at the lecture ready to see a new concept
 Arrive at the supplemental contact hour ready to practice that concept in facilitated group work
 Approach their homework ready to work independently
 Return to the lecture ready for an expansion on that concept, the introduction of a new concept, or an assessment of retention
This flow addresses the first two learning obstacles: weak prerequisite knowledge and passivity in learning.
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For some high school students, math is the bane of their existence. Others love numbers, logic, and how they can apply mathematical concepts to the real world.
But no matter which category you fall into, youll have to deal with math classes in high school. That said, not everyone has to take all the math classes available or reach the same levels.
Which math classes are common in high school curricula? And which ones do you actually need to take? Keep reading to find out.
How To Use This List
This list was created by researching the classes offered at numerous high schools, both public and private, across the country. Classes are alphabetically organized by subject. While there is a separate section for AP classes at the bottom of the list, other varying levels of difficulty for the same class, such as “honors” or “introductory”, were not included in order to make reading the list easier.
This list’s purpose is to show you all the possible class options you may have as a high school student. You can use it as a starting point for doing a more indepth study of your own school’s course offerings.
Read through the list below, making note of any courses that you may want to take in the future, then look to see if your school offers them. To find out which classes your own high school offers, look through your school’s course catalog, check the school website, or speak with your academic adviser.
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Ways To Support Students While They Work On A Modeling Prompt

Coach them on ways to organize their work better.

Provide a template to help them organize their thinking. Over time, some groups may transition away from needing to use a template.

Remind them of analog and digital tools that are available to them.

When students get stuck or neglect an important aspect of the work, ask them a question to help them engage more fully in part of the modeling cycle. For example:

What quantities are important? Which ones change and which ones stay the same?

What information do you know? What information would it be nice to know? How could you get that information? What reasonable assumption could you make?

What pictures, diagrams, graphs, or equations might help people understand the relationships between the quantities?

How are you describing the situation mathematically? Where does your solution come from?

Under what conditions does your model work? When might it not work?

How could you make your model better? How could you make your model more useful under more conditions?

What parts of your solution might be confusing to someone reading it? How could you make it more clear?
High School Precalculus Curriculum
With an emphasis on function families and their representations, Time4Learnings Precalculus course is a thoughtful introduction to advanced studies leading to calculus. The course briefly reviews linear equations, inequalities, and systems and moves purposefully into the study of functions. Students then discover the nature of graphs and deepen their understanding of polynomial, rational, exponential, and logarithmic functions. The course concludes with a short study of probability and statistics. Interactive math labs provide handson application of concepts using stateoftheart online tools.
Learn more about the high school precalculus curriculum.
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Organizing Students Into Teams Or Groups

Mathematical modeling is not a solitary activity. It works best when students have support from each other and their teacher.

Working with a team can make it possible to complete the work in a finite amount of class time. For example, the team may decide it wants to vary one element of the prompt, and compute the output for each variation. What would be many tedious calculations for one person could be only a few calculations for each team member.

The members of good modeling groups bring a diverse set of skills and points of view. Scramble the members of modeling teams often, so that students have opportunities to play different roles.
Csun Implementation And Results
In 2008, this course model was implemented at CSUN in the gateway Math 103 course. The results have been dramatic, essentially reversing the downward trend in student success:
 Prior to implementation, less than 34 percent of students received a final grade of C or better.
 Postimplementation, more than 66 percent of students achieve a C or better in Math 103.
These results have been sustained over seven semesters. Figure 2 compares the histograms of the grades on the final given in the semester before implementing the model in fall 2007 and the one after in spring 2008 . Note that the final each term is common to all sections of Math 103 and is collaboratively graded by the instructors and TAs. The exam varies somewhat from term to term, but not so significantly that comparisons cannot be made. The xaxis shows different percent ranges on the final exam, while the yaxis shows the percent of students in that range. For example, the height of a bar over 75 shows the percentage of students in each term that achieved 7580 percent on the final. As the graph shows, there was a vast improvement in both the students’ average scores and the distribution of the grades.
Figure 2. Final Exam Results Before and After Model Implementation
Figure 3. Students in the Computer Lab at CSUN
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When To Use Mathematical Modeling Prompts
A component of this is mathematical modeling prompts. Prompts include multiple versions of a task , sample solutions, instructions to teachers for launching the prompt in class and supporting students with that particular prompt, and an an analysis of each version showing how much of a lift the prompt is along several dimensions of mathematical modeling. A mathematical modeling prompt could be done as a classroom lesson or given as a project. This is a choice made by the teacher.
A mathematical modeling prompt done as a classroom lesson could take one day of instruction or more than one day, depending on how much of the modeling cycle students are expected to engage in, how extensively they are expected to revise their model, and how elaborate the reporting requirements are.
A mathematical modeling prompt done as a project could span several days or weeks. The project is assigned and students work on it in the background while daily math lessons continue to be conducted. This structure has the advantage of giving students extended time for more complex modeling prompts that would not be feasible to complete in one class period and affords more time for iterations on the model and cycles of feedback.
Modeling prompts dont necessarily need to involve the same math as the current unit of study. As such, the prompts can be given at any time as long as students have the background to construct a reasonable model.