Using high school algebra for a natural approach to derivatives and continuity

2018 ◽  
Vol 102 (555) ◽  
pp. 435-446
Author(s):  
R. Michael Range
Keyword(s):  
High School ◽  
Linear Equation ◽  
Quadratic Equation ◽  
Complex Numbers ◽  
New Techniques ◽  
Real Roots ◽  
High School Algebra ◽  
Natural Approach ◽  
Central Topic

The quadratic equation is a central topic in high school algebra. It provides the simplest generalisation of the familiar linear equation, and finding its roots introduces students to a non-trivial problem that requires the application of new techniques, such as completing the square and/or factorisation into linear factors involving the roots. It also introduces the student to the phenomenon of repeated roots, which opens the door to a discussion of multiplicities of roots. Furthermore, it naturally exposes the student to the case where the equation has no real roots, a phenomenon that could also be used to introduce the student to complex numbers.

The probability certain random quadratics have real roots

2021 ◽  
Vol 105 (564) ◽  
pp. 410-415
Author(s):  
Chris Boucher
Keyword(s):  
High School ◽  
Real Roots ◽  
High School Algebra ◽  
Integer Coefficients

Early in high school algebra, quadratics chosen as examples by teachers and textbooks alike tend to have integer coefficients and to factorise over the integers. This can give the misleading impression that such quadratics are the norm. As students progress into calculus and begin regularly seeing quadratics that are not as ‘nice’, we hope they become disabused of this notion. Indeed, even if the coefficients of the quadratic are integers, the probability that the quadratic factorises over the integers tends to zero as the range from which the integers are drawn grows (see [1]). But what if we ask about a behaviour less restrictive than factorising, say merely having real roots? This is the problem that concerns this Article.

Teaching Materials of Problem-Based Linear Equation System with Two-Variables for Eighth-Grade Students in Junior High School

2021 ◽  
Vol 1 (1) ◽  
pp. 119-123
Author(s):  
Nurhayati Abbas ◽  
Nancy Katili ◽  
Dwi Hardianty Djoyosuroto
Keyword(s):  
High School ◽  
Linear Equation ◽  
Junior High School ◽  
Junior High ◽  
Equation System ◽  
Eighth Grade ◽  
Teaching Material ◽  
Teaching Materials ◽  
Linear Equation System ◽  
Eighth Grade Students

This research is motivated by the lack of mathematics teaching materials that can make students learn on their own. The teaching material can be created by teachers as they are the ones who possess the knowledge about their students’ characteristics. Further, learning materials are a set of materials (information, tools, or texts) that can aid teachers and students to carry out the learning process. The two-variable linear equation system (SPLDV) is one of the mathematics materials taught to eighth-grade students of junior high school; it contains problems related to daily life. However, it is found that this material is still difficult to master by most students. Therefore, it is necessary to develop the SPLDV teaching materials that can help students learn and solve problems as well as be used as examples by teachers in developing other materials. This research aimed to make problem-based SPLDV teaching materials. The research method refers to the Four-D Model by Thiagarajan, Semmel, and Semmel (1974). It consisted of defining, designing, developing, and disseminating. The results showed that problem-based SPLDV teaching materials could be used in learning activities as the students and the teachers had shown their positive responses after going through expert assessments. This study also suggested that the teachers use this teaching material and adopt teaching materials for other similar materials.

New techniques and completeness results for preferential structures

2000 ◽  
Vol 65 (2) ◽  
pp. 719-746 ◽  
Author(s):  
Karl Schlechta
Keyword(s):  
Preference Relation ◽  
Algebraic Approach ◽  
Theory Revision ◽  
Main Interest ◽  
New Techniques ◽  
Natural Approach

AbstractPreferential structures are probably the best examined semantics for nonmonotonic and deontic logics: in a wider sense, they also provide semantical approaches to theory revision and update, and other fields where a preference relation between models is a natural approach. They have been widely used to differentiate the various systems of such logics, and their construction is one of the main subjects in the formal investigation of these logics. We introduce new techniques to construct preferential structures for completeness proofs. Since our main interest is to provide general techniques, which can be applied in various situations and for various base logics (propositional and other), we take a purely algebraic approach, which can be translated into logics by easy lemmata, in particular, we give a clean construction via indexing by trees for transitive structures, this allows us to simplify the proofs of earlier work by the author, and to extend the results given there.

Preparing solidly for algebra… using prime numbers

1964 ◽  
Vol 11 (6) ◽  
pp. 418-420
Author(s):  
W.A. Leonard
Keyword(s):  
High School ◽  
Elementary School ◽  
Prime Numbers ◽  
Statistical Evidence ◽  
Fatality Rate ◽  
Social Implication ◽  
High School Algebra ◽  
High Fatality Rate ◽  
Ways Of Thinking ◽  
First Time

It was once suggested that beginning the study of freshman high school algebra is not totally unlike embarking upon the sea of matrimony. Both are milestones in one's life, involving opportunities for excitement, adventure, growth, and rich and lasting reward; but, nonetheless, both tend to necessitate some different approaches to problems, some new ways of thinking, and certainly, in more than a few cases, some amount of adjustment. Without intending to attach social implication to algebra, we as teachers cannot fail to recognize that the first exposure to high school algebra can be a foreboding experience to many elementary school graduates, particularly in the light of the statistical evidence for the high fatality rate of students taking algebra for the first time.

“Plus” Work for “Plus” Pupils

1959 ◽  
Vol 6 (5) ◽  
pp. 251-256
Author(s):  
Eunice Lewis ◽  
Ernest C. Plath
Keyword(s):  
High School ◽  
Binary System ◽  
Sixth Grade ◽  
Number System ◽  
College Of Education ◽  
Base Number ◽  
High School Algebra ◽  
Second Year ◽  
The University

One plus one equals “10” for the members of a special arithmetic class at the University School, College of Education, Norman, Oklahoma. Of course, the members of this class were working with a number system of base two, commonly referred to as the binary system. Students also readily stated that three plus three equals “12” if the base is four. Changing the base number was not only fascinating to these highly talented fifth and sixth grade youngsters, but also provided a launching platform for the development of complicated formulas (patterns to them) which are normally developed in a second year high-school algebra course.

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