# Integration of Physics and Mathematics in STEM Education: Use of Modeling

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## Abstract

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## 1. Introduction

## 2. Framework

#### 2.1. Active Learning and STEM Education

#### 2.2. Modeling Activities and Real-World Scenarios

## 3. Methodology

#### 3.1. The Modeling Experience

#### 3.2. Data Collection

#### 3.3. Data Analysis

- -
- Angle: This involves disregarding the directional aspect of velocity, particularly overlooking that the ball’s initial velocity begins at a particular angle when it descends in the air.
- -
- Acceleration: This involves computing the ball’s acceleration along the ramp (1D).
- -
- Quadratic model: The initial stage involves a consistent one-dimensional (1D) acceleration, which is described by a quadratic model.
- -
- Final/initial velocity: Identifying that the final velocity at the end of the ramp is the initial velocity for the free ball motion.
- -
- Quadratic+linear model: The second stage comprises constant acceleration within a two-dimensional (2D) framework, employing a linear model along the horizontal axis and a quadratic model along the vertical axis.
- -
- Hit: Whether the ball enters the cup or not.

## 4. Results

## 5. Discussion

#### 5.1. Interpretation of Results

#### 5.2. Importance of Modeling Instruction

#### 5.3. Integration of Physics and Mathematics

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Model development sequence embedded in modeling instruction (adapted from [24]).

**Figure 2.**Drawing of the situation indicating the measurements in blue and the unknowns in red. (

**a**) realistic representation and (

**b**) a triangle used to calculate the ramp’s angle.

**Figure 3.**During the Model-Application Activity (MAA), students worked in groups and gathered around the ramp to test their values (d and D).

**Figure 4.**Two whiteboards of students’ groups showing their work in solving the Model-Application Activity.

Model | Physics Objective | Mathematics Objective |
---|---|---|

1D motion | Build how position, velocity, and acceleration graphs relate, and build a specific model for constant acceleration. | Introduce different types of functions and how they relate to the graphical, equation, and verbal ways of representing them. |

2D motion | Realize that displacement, velocity, and acceleration are vectors. Generalizing the previous model from 1D to 3D. Students should now be able to explain how things move given certain conditions. This cycle ends in the ball-in-cup problem. | Functions |

Energy | Through analysis of the bounce of a ball, students can find that energy is conserved and will be the first reason things move. | Euler |

Forces Part 1 | Using force sensors builds the relationships between acceleration and forces—a second way of understanding why things move. Students should be able to explain the movement of individual objects. | Derivatives |

Forces Part 2 | Adding a second object to the interactions builds how forces are applied between objects. Now, students can analyze more than one object interacting. All of Newton’s three laws have been covered. | Derivatives and their applications. |

Other Forces | Other forces are analyzed; the force on a spring is the most common. The student performs different measurements in building how they relate to other variables. | It is a great place to play with the sine function. Both how they work and their derivative and integral relations. They can play with a mass-spring oscillation system. |

Circular motion | Use various experiments to understand how the direction of the forces applied to an object can create specific kinds of motion. Build the particular case of how radius and velocity relate to the acceleration towards the center. | Integrals |

Momentum | A new unit that can easily explain what happens when a collision is introduced. Notice it is a vector and should be treated as such. | Integrals |

Rotational motion | Students realize that all we have built until now is in a Cartesian system. Use angle instead of position to relate it to a circular system and prove that everything is the same. Introduce rotational inertia, rotational forces (torque), rotational energy, and rotational momentum. Prove relationships are still the same as in a Cartesian system. | Integrals |

**Table 2.**Results for the 2D cup model. Interpretation and values were taken from students’ whiteboards.

Team | Angle | Accel. | Quadratic Model | Final/Initial Velocity | Quadratic+Linear Model | Hit |
---|---|---|---|---|---|---|

1 | Yes | No | No, Units | No, Units | No, Units | No |

2 | Yes | Yes | Yes | Yes | Yes | Yes |

3 | Yes | Yes | Yes, Units | Yes, Units | No, Units | No |

4 | Yes | No | No, Units | No, Units | No | No |

5 | Yes | Yes | Yes | Yes | Yes | Yes |

6 | Yes | Yes | Yes, Units | Yes, Units | Yes, Units | No |

7 | Yes | Yes | Yes | Yes | Yes | Yes |

8 | Yes | Yes | Yes, Units | Yes, Units | Yes, Units | No |

9 | Yes | Yes | Yes | No, Units | No | No |

10 | Yes | Yes | No | No | No | No |

11 | Yes | Yes | Yes | Yes | Quadratic formula error | No |

12 | Yes | Yes | No | No | No | No |

13 | Yes | Yes | Yes | Yes | No | No |

14 | Yes | Yes | Yes | Yes | Yes | Yes |

15 | Yes | No | No | No | No | Yes |

16 | Yes | Yes | Yes | Yes | No | No |

17 | Yes | N/A | N/A | N/A | N/A | No |

18 | Yes | N/A | N/A | N/A | N/A | No |

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**MDPI and ACS Style**

Dominguez, A.; De la Garza, J.; Quezada-Espinoza, M.; Zavala, G.
Integration of Physics and Mathematics in STEM Education: Use of Modeling. *Educ. Sci.* **2024**, *14*, 20.
https://doi.org/10.3390/educsci14010020

**AMA Style**

Dominguez A, De la Garza J, Quezada-Espinoza M, Zavala G.
Integration of Physics and Mathematics in STEM Education: Use of Modeling. *Education Sciences*. 2024; 14(1):20.
https://doi.org/10.3390/educsci14010020

**Chicago/Turabian Style**

Dominguez, Angeles, Jorge De la Garza, Monica Quezada-Espinoza, and Genaro Zavala.
2024. "Integration of Physics and Mathematics in STEM Education: Use of Modeling" *Education Sciences* 14, no. 1: 20.
https://doi.org/10.3390/educsci14010020