A Pattern in Number Theory: Example→ Generalization→ Proof

1971 ◽  
Vol 64 (7) ◽  
pp. 661-664
Author(s):  
David R. Duncan ◽  
Bonnie H. Litwiller
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
Mathematical Induction ◽  
Whole Number

The study of patterns is an integral part of the study of mathematics. As we teach mathematics, we must point out how to search for patterns and how patterns may aid us in problem solving. The following problem is one that combines patterns, ideas from number theory, and mathematical induction: “Prove that it is possible to pay, without requiring change, any whole number of rubles (greater than 7) with banknotes of value 3 rubles and 5 rubles” (Sominskii 1964, p. 19).

scholarly journals Analisis Kesulitan Soal Pemecahan Masalah Pada Buku Sekolah Elektronik (BSE) Pelajaran Matematika Kurikulum 2013 revisi 2017

2021 ◽  
Vol 5 (2) ◽  
pp. 455-468
Author(s):  
Yulia Rahayu ◽  
Ali Umar ◽  
Rosliana Harapan
Keyword(s):  
Problem Solving ◽  
Difficulty Level ◽  
Problem Type ◽  
Literature Study ◽  
Curriculum Revision ◽  
Analysis Technique ◽  
Whole Number ◽  
Mathematical Problems ◽  
Collection Methods

This study aims to analyze and describe the level of difficulty of solving mathematical problems in terms of the components of the problem: type of problem, type of number, type of operation, many operations, many questions, sufficiency of data, and similarity with previous questions and difficulty level of problem solving questions in the Electronic School Book (BSE) Mathematics for Senior High School (SMA) class X 2013 curriculum revision of 2017. This research uses literature review or literature study with a descriptive qualitative approach. Data collection methods used were the determination of mathematics teaching materials/textbooks; determination of problem-solving items contained in the competency test. The data analysis technique uses the stages of data reduction, data presentation, and conclusion. The results showed: 1) There are 18% problem solving questions or 20 questions out of 109 total questions; 2) Type of problem solving 55% routine and 45% nonroutine; 2) the dominant number type of whole number is 60%; 3) More types of operations use multiplication operations as much as 90%; 4) Many dominant operations use more than one operation as much as 100%; 5) Many dominant questions use one question as much as 70%; 6) Has sufficient complete data as much as 75%; 7) As many as 60% of the questions were not similar to the previous questions. So that the problem solving questions in the School Electronic Book (BSE) class X SMA are categorized as having an Easy difficulty level.

Even Perfect Numbers—An Update

1981 ◽  
Vol 74 (6) ◽  
pp. 460-463
Author(s):  
Stanley J. Bezuszka
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
History Of Mathematics ◽  
Independent Study ◽  
History Of ◽  
Perfect Numbers ◽  
Excellent Resource

Do you have students who are computer buffs, always looking for a new problem to program efficiently? Do you have students who do independent study projects? If so, motivate them with this topic that is rich in the history of mathematics and number theory—perfect numbers. They provide an excellent resource for theoretical as well as computerized problem solving.

Technology and Reasoning in Algebra and Geometry

1996 ◽  
Vol 89 (2) ◽  
pp. 138-142
Author(s):  
Daniel B. Hirschhorn ◽  
Denisse R. Thompson
Keyword(s):  
Number Theory ◽  
Mathematical Induction ◽  
Educational Progress ◽  
National Assessment ◽  
Trigonometric Identity ◽  
Reasoning Skills ◽  
Geometry Teachers ◽  
Long Course

If one topic is likely to be stressed by algebra and geometry teachers, it is reasoning. In algebra classes, students are constantly being asked to show their work and justify their simplifications, often without formal connection to proof concepts or the proof process. In geometry classes, students are expected to learn how to write simple proofs. However, evidence shows that students are not learning these reasoning skills. In the 1985–86 National Assessment of Educational Progress, Silver and Carpenter (1989, 18) found that “many eleventhgrade students are confused about the fundamental distinctions among mathematical demonstrations, assumptions, and proofs.” Most students thought a theorem was a demonstration or an assumption. Senk (1985) found that only about 30 percent of students mastered proof wTiting in geometry, despite being enrolled in a year-long course emphasizing proof. Thompson (1992) found that roughly 60 percent of precalculus students were successful at trigonometric-identity proofs, more than 30 percent could complete number-theory proofs dealing with divisibility, and less than 20 percent could handle indirect arguments or proof by mathematical induction.

scholarly journals PENERAPAN MODEL PEMBELAJARAN TERPADU TIPE NESTED DALAM PEMBELAJARAN PEMECAHAN MASALAH MATEMATIKA PADA MATERI INDUKSI MATEMATIKA

2020 ◽  
Vol 6 (2) ◽  
pp. 113
Author(s):  
Aan Armini
Keyword(s):  
Problem Solving ◽  
Positive Attitude ◽  
Mathematical Problem ◽  
Learning Model ◽  
Mathematical Induction ◽  
Mathematical Problem Solving ◽  
Integrated Learning ◽  
School Year ◽  
Post Test ◽  
Students Attitudes

Through classroom action research, researchers seek to improve students 'mathematical problem solving abilities on mathematical induction material by using nested integrated learning methods and explore students' attitudes towards the application of the given learning model. The study was conducted during two cycles which included four stages of learning, namely: planning, implementation, observation, and reflection. The study was conducted in the odd semester of the 2019/2020 school year involving 34 students of class XI MIPA-2 SMAN 1 Garawangi, Kuningan. There are 2 types of research instruments used, namely tests and questionnaires. Based on the test results, it can be seen that in the post-test, the N-Gain index obtained was 0.61 (moderate). Meanwhile, based on the results of the questionnaire, it can be seen that 100% of students show a positive attitude. Thus, it can be concluded that the nested type integrated learning model can improve students' mathematical problem solving abilities on mathematical induction material and get a good appreciation with the presence of a positive attitude shown by all participants.

Teaching Teachers to Teach Mathematics

1996 ◽  
Vol 178 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Rika Spungin
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
Prospective Teachers ◽  
Problem Solving Strategies

Group investigations from the areas of number theory, probability, and geometry. are presented and discussed. By working in groups, sharing ideas, and making and testing conjectures, prospective teachers gain confidence in their own ability to do mathematics and develop a variety of useful problem-solving strategies.

Polygons, Stars, Circles, and Logo

1986 ◽  
Vol 33 (9) ◽  
pp. 6-11
Author(s):  
Bill Craig
Keyword(s):  
Middle School ◽  
Elementary School ◽  
Number Theory ◽  
Problem Solving ◽  
Elementary School Students ◽  
Upper Elementary ◽  
School Students ◽  
Know How ◽  
Elementary And Middle School ◽  
Mathematical Topic

Many teacher are excited about the potential uses of Logo with elementary school students. The language give students access to mathematical topic they have not previouly explored. The following activitie uae Logo in the study of geometry, number theory, and problem solving. The activities assume that tudents are familiar with turtlegraphic commands (FORWARD, BACK, RIGHT, LEFT) and know how to define procedures. The activitie are designed for students in the upper elementary and middle school grades. The star procedure and explorations are adapted from Discovering Apple Logo by David Thornburg. The book contains excellent ideas for the use of Logo as a tool for mathematical explorations. See the Bibliography for additional resources.

Tacoma Shuffle

1998 ◽  
Vol 91 (3) ◽  
pp. 212-216
Author(s):  
Lyman S. Holden ◽  
Loyce K. Holden
Keyword(s):  
Problem Solving ◽  
Mathematical Induction ◽  
Computer Application ◽  
Trial And Error ◽  
C Language ◽  
Key Concepts ◽  
Proof By Induction ◽  
Iterative Functions

The key concepts discussed in this article include problem-solving activities, mathematical induction, proof by induction, and use of the phrase “without loss of generality.” Several problem-solving tools are illustrated, such as trial and error, working backward, and seeing patterns. The computer application illustrates recursive and iterative functions using C language.

Counting Triples, Triangles, and Acute Triangles

1999 ◽  
Vol 92 (7) ◽  
pp. 612-619
Author(s):  
Ruth McClintock
Keyword(s):  
Problem Solving ◽  
Difference Equation ◽  
Mathematical Induction ◽  
Modular Arithmetic ◽  
Advanced Students ◽  
Greatest Integer Function ◽  
First Time

Activities involving counting triples, triangles, and acute triangles enrich the curriculum with excursions into modular arithmetic, the greatest-integer function, and summation notation. In addition, more advanced students can apply difference-equation techniques to find closed forms and can use mathematical induction to prove the formulas. Students may be learning about these topics for the first time, or they may be reviewing familiar ideas in different problem-solving contexts. In either situation, personal arsenals of problem-attacking skills are strengthened.

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