[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-278100-105":53,"doc-detail-278100-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","teaching-locus-with-a-conserved-property-integrating-mathematical-tools-and-dynamic-geometric-software","Teaching Locus with a Conserved Property - Integrating Mathematical Tools and Dynamic Geometric Software","","Investigative dynamic geometry tasks explore mathematical loci by integrating software (DGS) with mathematical reasoning for proving generated configurations. Additional conditions—such as ellipse, parabola, and circle—produce new loci similar in form to the originals, while learners derive parameter relationships between original and new loci. A general mathematical explanation connects the “surprising” results from the investigations. The paper argues that integrating DGS into mathematics instruction improves teaching and learning, supporting deeper concept understanding through conserved properties and generalization.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/teaching-locus-with-a-conserved-property-integrating-mathematical-tools-and-dynamic-geometric-software/278100/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/teaching-locus-with-a-conserved-property-integrating-mathematical-tools-and-dynamic-geometric-software/278100.png","ImageObject",442,249,{"name":88,"@type":89},"8796093062539","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-09-15",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What role does dynamic geometric software play in the investigation of loci?","Question",{"text":109,"@type":110},"Dynamic geometry software is integrated with mathematics to generate and explore loci, then to support proving obtained figures rather than relying on screen output as proof.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How are new loci produced beyond essential loci?",{"text":114,"@type":110},"By adding mathematical conditions such as ellipse, parabola, and circle, learners investigate how these constraints lead to loci that are similar in shape to the original loci.",{"name":116,"@type":107,"acceptedAnswer":117},"Why is pedagogical content knowledge (PCK) relevant to this teaching approach?",{"text":118,"@type":110},"PCK describes what teachers know about representation, explanations, misconceptions, and strategies for making learning easier, and it supports planning and evaluating mathematical tasks integrated into instruction.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},278100,1791733321,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":60,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":141},8796093062539,"Teaching locus with a conserved property by integrating mathematical tools and dynamic geometric software  \nMoshe Stupel, Ruti Segal & Victor Oxman  \nShaanan College, Haifa, Israel  \nstupel@bezeqint  . net  \nrutisegal@gmail  . com  \nvictor . oxman@gmail  . com  \nWe present investigative  \nof dynamic geometry  \ntasks that concern loci, which integrate the use software (DGS) with mathematics for proving  \nthe obtained figures. Additional conditions were added to the loci: ellipse, parabola and circle, which result in the emergence of new loci, similar in form to the original loci. The mathematical relation between the parameters of the original and new loci was found by the learners. A mathematical explanation for the general case, using the ‘surprising’ results obtained in the investigative tasks, is presented. Integrating DGS in mathematics instruction fosters an improved teaching and learning process.  \nIntroduction  \nA conserving property maybe used to solve different problems in mathematics and the identification of such a property leads to a complete and deep understanding of mathematical concepts and the relationships between them. The locus, according to this definition, conserves a certain property and can therefore be used to prove a wide variety of problems and even serve in the execution of construction problems using a straightedge and a compass, as has been done in mathematics since antiquity.  \nLoci can be divided into two groups:  \n1. Essential loci, such as: the straight line, parallel lines, the mid perpendicular of a segment, an angle bisector, the circle, the parabola, the ellipse and the hyperbola.  \n2. Loci which are a result of the essential loci or loci obtained as a result of satisfying several mathematical conditions.  \nAnalytical geometry is one prominent field in mathematics in which extensive use is made of loci.  \nIn most colleges for teacher training, prospective teachers of the highschool (ages 12–17 years old) stream in mathematics are taught four sequential courses, which deal with loci.  \nAustralian Senior Mathemat ics Journal vo l. .30 no 1  \n25  \nStupel, Sega l & Oxman  \n26  \nAustralian Senior Mathematics Journal vo l . 30 no . 1  \nThe first course is the elements of Euclidean geometry. The second course is the principles of analytical geometry which begins with the straight line and ends with the circle. The third course is integration of computer technology in the teaching of mathematics where the students acquire the skills of using different computerised technological tools. The fourth course is advanced analytical geometry, which begins with the parabola, the ellipse and the hyperbola and ends with unique subjects, and deals with loci, making maximum use of the knowledge acquired in the three previous courses, and in particular, the use of computer technology. Advanced analytic geometry is intended to demonstrate the importance of the locus as an important tool in the mathematical toolbox of solving different problems where identification of conserved properties allows one to obtain a mathematical generalisation. In this subject the loci are a basis for additional tasks which include more requirements, and allow one to obtain new loci based on the given loci.  \nSometimes an introduction of additional requirements brings about formation of a locus whose shape is the same as that of the original locus, and other times a different locus is formed. The present paper deals with obtaining loci which are similar in shape to the original locus.  \nIn some cases mathematical conditions are given and the student is required to discover what essential locus is obtained.  \nNaturally in all cases the students were required to prepare formal mathematical proofs for the locus shape obtained. It should be emphasised that the students must know that what is seen on the computer screen is not mathematical proof, but only a basis for speculation.  \nIn some tasks the students found several proofs as a result","cbCaigz7x4owtsO7","https://ap.wps.com/l/cbCaigz7x4owtsO7","pdf",1029858,20,"English","# Introduction\n## Conserved property and loci\n## Essential vs. constructed loci\n# Teaching through dynamic geometry tasks\n## Additional conditions and new loci\n## Proof vs. computer visualization\n## Multiple mathematical approaches\n# Development of pedagogical content knowledge\n## PCK and MKT\n## Teacher knowledge and task integration","[{\"question\":\"What role does dynamic geometric software play in the investigation of loci?\",\"answer\":\"Dynamic geometry software is integrated with mathematics to generate and explore loci, then to support proving obtained figures rather than relying on screen output as proof.\"},{\"question\":\"How are new loci produced beyond essential loci?\",\"answer\":\"By adding mathematical conditions such as ellipse, parabola, and circle, learners investigate how these constraints lead to loci that are similar in shape to the original loci.\"},{\"question\":\"Why is pedagogical content knowledge (PCK) relevant to this teaching approach?\",\"answer\":\"PCK describes what teachers know about representation, explanations, misconceptions, and strategies for making learning easier, and it supports planning and evaluating mathematical tasks integrated into instruction.\"}]","Teaching Locus with a Conserved Property - Integrating Mathematical Tools and Dynamic Geometric Software | PDF",1789506981,7]