linear and quadratic functions

Journal on Mathematics Education

Mathematical representation has an essential role in solving mathematical problems. However, there are still many mathematics education students who have difficulty in representing ill-structured problems. Even though the ill-structured-problem-solving tasks designed to help mathematics education students understand the relevance and meaningfulness of what they learn, they also are connected with their prior knowledge. The focus of this research is exploring the used of mathematical representations in solving ill-structured problems involving quadratic functions. The topic of quadratic functions is considered necessary in mathematics teaching and learning in higher education.…

2016

Teaching mathematics in a high school and in a community college in the New York City, I have noticed that students grasp mathematical concepts more quickly and understand them better when a connection between the concept and the world of business is established. Students were visibly more interested and involved when business applications were presented first, laying the groundwork for technological applications and for the introduction of further theoretical concepts. This observation, well known to mathematics instructors, deserves further research. This paper aims to show how the business approach can be used in teaching quadratic functions and parabolas: a topic encountered at every level of mathematics and one with a wide range of applications. It will be shown how business information may be presented in a classroom to improve students’ understanding of the topic. The paper assumes that students have had some practice in quadratic functions and graphing parabolas.

Lesson Plan Quadratic Function

College Mathematics Journal, 1995

This paper describes the trial of a unit of work on linear and quadratic graphing with six year 10 classes. Two treatments were developed. The computer treatment made use of the ANUGraph software package, while the calculator treatment paralleled the computer treatment but used a combination of previously prepared graphs and graphs constructed by the student with the aid of a calculator. The emphasis in both treatments was on the interpretation of graphs related to real situations. Comparisons between pre-test and post-test results and interviews with twelve students showed that students learnt to handle the software proficiently, and that both groups improved on most of the topics taught. However, the calculator group seemed to be advantaged by practising plotting of points by hand. Implications for future work are discussed. Most secondary schools are now equipped with computers that can run graphing and spreadsheet software, although they are still not frequently used for mathematics teaching. In Asp, Dowsey and Stacey (1992) we described a trial of a teaching unit using spreadsheets to assist students in constructing meaning for the important concepts of variable, expression, equation and solution. This teaching experiment gave an indication of which ideas about equations become easier with appropriate computer technology and what difficulties, both conceptually and in terms of classroom management, result from the use of this technology. In this paper we describe results from a trial of a teaching unit on linear and quadratic functions and their graphs, a topic which seems similarly suited to the use of computers. Frequently all aspects of a complex mathematical idea cannot be expressed with a single representational system. The idea may require multiple, linked representations for its full expression and these different representations may aid the learner's understanding of the idea. Kaput (1992) sees the ability to make translations from one representation of a function to another as a particularly important aspect of mathematical thinking which may be enhanced by technology. The convenient access provided by graphing software to numerical and graphical representations of a variety of functions may assist students to develop a broader and deeper understanding of the function concept.

This note briefly explains quadratics suited for Cambridge AS and A-level mathematics, Cambridge IGCSE additional mathematics, and analysis and approaches mathematics for IB Diploma Programme.

International Journal of Science and Mathematics Education, 2015

This study describes a comparison of how worked examples in selected textbooks from England and Shanghai presented possible learning trajectories towards understanding linear function. Six selected English textbooks and one Shanghai compulsory textbook were analysed with regards to the understanding required for pure mathematics knowledge in linear function. Understanding was defined as being at five levels: Dependent Relationship, Connecting Representations, Local Properties Noticing, Object Analysis, and Inventising. These levels were developed by examining the most prominent theories from the existing literature on understanding function. Findings suggested that the English textbooks constrained the structural aspect of understanding linear function due to a point-to-point view of function, while the Shanghai textbook which focussed on a variable view of function overemphasised the algebraic approach. The discussion explored the drawbacks to each approach and what teachers or textbook writers could do to balance these two approaches in order to facilitate students' understanding towards a structural view of linear function.

Eurasia Journal of Mathematics, Science and Technology Education, 2020

The article aims to determine the academic performance and errors in the resolution of types of problems of application of the quadratic function, of high school students from the Los Lagos Region and Los Rios Region in Chile. The approach is qualitative and descriptive with case studies. A math test with open response problems and an opinion questionnaire were developed and applied. Through the results, the highest academic performance is evidenced in the routine problems of purely mathematical context and fantasist context, but with difficulty in the resolution of non-routine problems. In addition, errors originating in affective and emotional attitudes associated with blockages at the time of initiating the resolution, forgetfulness at the time of posing the quadratic function, prevail over cognitive errors originating in an obstacle and errors originating in the absence of meaning.

1997

International Journal of Advanced Science and Engineering, 2023

The concept of function is one of the fundamental mathematical concepts, very important within mathematics itself as well as in the application of mathematics. Functions are an essential element of mathematical structuring and modeling of problems (e.g.in algebraic structures), as well as a means of comparing structures thus obtained (eg homomorphisms of structures). A mathematical function is a rule that gives the value of the dependent variable corresponding to certain values of one or more independent variables. A function can be represented in several ways, such as a table, formula, or graph. Apart from isolated points, the mathematical functions found in physical chemistry are single-valued. Apart from isolated points, the mathematical functions that occur in physical chemistry are continuous.

2022

This work aims to present the results of an investigation in the teaching of quadratic function with the help of the PhET Colorado simulator, analyzed from the perspective of the theory of figural concepts in the context of hybrid teaching, using the teaching methodology flipped classroom. The research methodology used was the case study, which was developed with a group of 45 high school students from a Brazilian public school. The proposed activity was developed from the simulation called "graphing quadratics", available in the PhET, and was developed in two stages, one in a virtual way and the other in person. The results show us the need to explore the study of the quadratic function using technology from a more dynamic perspective. We reinforce the importance of the manipulation performed in the simulator to understand the relationship between the coefficients a, b and c of the function and the behavior of its graph, being a potential resource in the learning of this subject by the students.

JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES, 2022

The basic idea of a quadratic equation is one of the most important topics in algebra. The mathematical concept for the method of solution of a quadratic equation is dependent on the advancement of the theory of numbers. The author developed a new concept regarding the method of solution of the quadratic equation based on "Theory of Dynamics of Numbers". The author determined the inherent nature of one unknown quantity (say x) from the quadratic expression 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 of the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 by keeping the structure of the second-degree expression intact and then finding the solution of the quadratic equation using the novel concept of the Theory of Dynamics of Numbers. The author solved any quadratic equation in one unknown number (say x) of the quadratic equation in the form of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, whether the numerical value of the discriminant is 𝑏 2-4𝑎𝑐 ≥ 0 or 𝑏 2-4𝑎𝑐 < 0, is real numbers only without using any imaginary numbers. With these new inventive concepts, the author developed new theories in the theory of quadratic equation.

International Journal of Science and Mathematics Education, 2020

This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. In particular, we describe students' work on a Task designed to examine the connection between points on a graph and the equation of a line. The data consist of 63 responses to a written questionnaire and individual interviews with three participants. The results indicate that visual approaches impede students' solutions and point to incomplete connections between algebraic and graphical representation. While algebraic approaches point to various connections used in approaching the Task, students' ability to work with algebraic representation did not necessarily result in capitalizing on these connections. Furthermore, interpretations of the graph based on visual inspection appeared most useful when used in support of the algebraic approach.

ZDM – Mathematics Education

This study lies within the field of early-age algebraic thinking and focuses on describing the functional thinking exhibited by six sixth-graders (11- to 12-year-olds) enrolled in a curricular enhancement program. To accomplish the goals of this research, the structures the students established and the representations they used to express the generalization of the functional relationship were analyzed. A questionnaire was designed with three geometric tasks involving the use of continuous variables in quadratic functions. The students were asked to calculate the areas of certain figures for which some data were known, and subsequently to formulate the general rule. The results show that the participating students had difficulties expressing structures involving quadratic functions. However, they displayed the potential to use different types of representations to establish the functional relationship. The originality of this study lies in the differences observed in the process of g...