## Introduction

Events of instruction of a lesson are constructed based on the learning outcomes and objectives of the class. Learning outcomes are the quantifiable statements that express what learners should value, know, or do at the end of the lesson. The significance of providing these accounts at the beginning of the instruction is that they form the cornerstone of assessments and task design (Hashemi et al., 2020). They ensure students pay attention to what is vital by clarifying expectations. On the other hand, learning objectives organize the specific activities or topics that should be covered to achieve the overall outcome of the lesson. This assessment discusses the learning outcomes and performance objectives of the selected introduction to calculus lesson. The events of instruction are based on the constructivism learning theory, which is characterized by active learners and a student-centered approach to teaching.

## Learning Outcomes

The task analysis of the introduction to calculus lesson identified critical or cognitive thinking, technological skills, and visual and verbal communication as essential competencies in the class. The first learning outcome is that at the end of the course, the students should be able to state the fundamental calculus theorem, the formal description of the derivative of a function, and the roles of all symbols in the explanation (Rahman et al., 2020). Secondly, they should be able to compute the graphical, algebraic, and numerical limits of a function and apply the differentiability and continuity notions to transcendental and algebraic functions. Lastly, the learners are expected to understand and apply the definite integral principles to calculate the rate of change, total change of a function, and areas below or above curves. In other words, they should be able to use integration and differentiation to resolve real-world challenges.

## Performance Objectives

Students should demonstrate performance at the end of the lesson by explaining the relationship between integrals and derivatives using the fundamental calculus theorem. They should be able to use limits to establish whether a function is differentiable or continuous at a particular point (Rahman et al., 2020). Thirdly, the learners should be able to solve area and tangent problems using the principles of integrals, limits, and derivatives. Another learning objective that students should accomplish by the end of the class is to use differentiation concepts to differentiate transcendental and algebraic functions. Additionally, they should be able to draw graphs of the aforementioned functions bearing in mind the rules of differentiability at a specified point, continuity, and limits. Finally, learners should be able to identify the appropriate calculus techniques and principles to obtain mathematical models to find answers to real-world problems.

## Why the Selected Objectives Are Appropriate

Learning calculus is essential for science students because the topic can be hugely applied in various settings. Calculus principles are not limited to analysis and mathematics; they can be used in physics, economics, dynamic systems, and engineering. Therefore, the aforementioned learning and performance objectives are crucial because they help students master three essential tools: integrals, derivatives, and limits. The competencies can be utilized in solving application problems in many settings, including business environments. Through the performance objectives, learners will be able to compose detailed responses to functions using correct mathematical language. They will be able to identify and apply calculus to other subjects and fields where the course is significant.

Lastly, completing the learning objectives will enable students to generate answers to unfamiliar challenges in the world today. Applying critical thinking to determine solutions to advanced problems in the business world is the most crucial aspect of learning calculus. The learning outcomes can be achieved sequentially through describing fundamentals, synthesis, and mastery. Fundamentals are the building blocks of the lesson, and they often appear at the beginning of the class. They form the basis of more complex principles and are combined by breaking problems into small practical steps in the synthesis stage. Each division of the challenge is solved separately and conjoined to create one solution. A key component of synthesis is matching appropriate techniques to open-ended problems and considering the particular rules of the chosen principle. Eventually, students will master the content and be able to take a calculus question, sub-divide it into parts, craft a formula and execute and interpret results. Critical aspects of mastery include differentiating between competing principles, evaluating the effectiveness of functions in different contexts, and explaining results competently.

## Events Mapped to Each Objective

## How are Events Effective and Aligned

To attain the performance objectives, events of instructions that ensure each goal is achieved should be formulated. Educators will use lecture outlines to teach students the fundamental calculus theorem and the relationship between integrals and derivatives. The lectures will cover the definition of terms, concepts, problems, and practice questions (Stanberry, 2018). Handing out the outline in advance will guarantee that students will organize their notes well and correctly copy the problems and questions. The second objective of using limits to establish the differentiability of a function at a point can be instructed through group discussions. The educator can put students in groups depending on their capabilities and give them questions to solve. Discussing in teams will enable students to enhance their problem-solving and critical thinking skills. Additionally, it builds the confidence of students, making them express themselves better (Mingus & Koelling, 2021). Thirdly, for area and tangent problems, students will be taught to solve them using worked examples that show the step-by-step mechanism of solving a task. The method supports the initial acquisition of critical thinking skills by introducing a situation and demonstrating the steps to reaching a final solution.

Another specified learning objective was using differentiation techniques to differentiate transcendental and algebraic functions. Inquiry-based learning will accomplish the goal by connecting learners to real-world problems through high-level questioning and exploration. Educators will supplement group activities with assignments and independent research to ensure this outcome is reached. In addition, instructors will use technological tools (SimCalc software and a computer algebra system) to teach learners how to draw and interpret graphs (Tatar et al., 2020). The techniques support visualization as they allow symbolic and graphical computations of functions. Using this method will ensure students understand the relationship between functions and their derivatives and interpret graphical representations (Jafar et al., 2020). The last performance objective is identifying the best technique to develop an arithmetic model for answering situational problems. Students will develop this competency through problem-based learning, a student-centered approach that involves working in groups to find solutions to open-ended questions. In this approach, students are motivated to learn through encountering problems and discussing with others to find the appropriate mechanism to solve them. The practice will ensure learners deepen their understanding of the calculus concepts and how and why they are applied in a particular manner.

## Conclusion

To conclude, events of instruction are formulated based on performance objectives and their associated learning outcomes. Educators must include these aspects in their lesson plans to ensure the class content focuses on the expected learning outcomes. In the introduction to calculus lesson, the selected teaching techniques are mapped to the performance objectives to achieve all the learning goals. Subsequently, the learning objectives are matched to learning outcomes and the overall task to ensure students develop cognitive skills to know the appropriate principles to apply in specific open-ended and real-world problems. Through fulfilling the aforementioned objectives and outcomes, the overarching goal of the lesson will be achieved.

## References

Hashemi, N., Kashefi, H., & Abu, M. (2020). The Emphasis on Generalization Strategies in Teaching Integral: Calculus Lesson Plans. *Sains Humanika*, *12*(3), 35-43. Web.

Jafar, A. F., Rusli, R., Dinar, M., Irwan, I., & Hastuty, H. (2020). The effectiveness of video-assisted flipped classroom learning model implementation in integral calculus. *Journal of Applied Science, Engineering, Technology, and Education*, *2*(1), 97-103. Web.

Mingus, T., & Koelling, M. (2021). A collaborative approach to coordinating Calculus 1 to improve student outcomes. *PRIMUS*, *31*(3-5), 393-412. Web.

Rahman, M., Ling, L., & Yin, O. (2020). Augmented reality for learning calculus: a research framework of interactive learning system. In *Computational science and technology* (pp. 491-499). Springer, Singapore.

Stanberry, M. (2018). Active learning: A case study of student engagement in college calculus. *International Journal of Mathematical Education in Science and Technology*, *49*(6), 959-969. Web.

Tatar, D., Roschelle, J., & Hegedus, S. (2020). SimCalc: Democratizing access to advanced mathematics (1992–present). In *Historical instructional design cases* (pp. 283-314). Routledge.