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It explains how modular arithmetic uses remainders to create a low- to high-resolution pattern, then converts table values into angles and draws line segments on a transformed pixel canvas. The notes connect the resulting envelope curves to epicycloids such as cardioid, nephroid, and trefoil, and include investigation prompts plus parameterized-equation questions for loci.",{"@graph":69,"@context":121},[70,84,104],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/string-art-on-the-unit-circle-teacher-notes-answers/674763/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":98,"encodingFormat":97,"isAccessibleForFree":99,"interactionStatistic":100},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/string-art-on-the-unit-circle-teacher-notes-answers/674763.png","ImageObject",300,407,{"name":92,"@type":93},"River Wang","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04",true,{"@type":101,"interactionType":102,"userInteractionCount":4},"InteractionCounter",{"@type":103},"ViewAction",{"@type":105,"mainEntity":106},"FAQPage",[107,113,117],{"name":108,"@type":109,"acceptedAnswer":110},"How does modular arithmetic help generate the unit-circle string-art pattern?","Question",{"text":111,"@type":112},"It uses remainder (mod) to map multiplication results into a fixed range, avoiding problems when products reach values at or above 12. This mapping can be generated directly for low resolution or via a program for higher resolution patterns.","Answer",{"name":114,"@type":109,"acceptedAnswer":115},"What transformations are used to center and scale the diagram on TI-Nspire?",{"text":116,"@type":112},"Because the TI-Nspire drawing origin is at the top left, the notes apply a horizontal shift h := 159 and vertical shift v := 106 to center the figure, then use a dilation r := 100 to scale the unit circle appropriately.",{"name":118,"@type":109,"acceptedAnswer":119},"What locus and parameter equations result when the outer circle traces around the inner circle with R = r?",{"text":120,"@type":112},"With point P at the outer-circle center, P traces a circle with x(t)=4cos(t) and y(t)=4sin(t) (CCW) or y(t)=-4sin(t) (CW). For a point on the outer circle, the path is x(t)=4cos(t)+2cos(2t) and y(t)=4sin(t)+2sin(2t).","https://schema.org",{"og:url":83,"og:type":123,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":125,"canonical":83},"index,follow",{"doc_id":127,"site_id":62},674763,1793767402,{"code":4,"msg":5,"data":130},{"doc_id":127,"user_id":131,"nickname":92,"user_avatar":132,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":4,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":138,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":139,"faqs":140,"seo_title":141,"seo_description":67,"update_tm":128,"read_time":46},1099514067438,"https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542","String Art on the Unit Circle  \nTeacher Notes & Answers      \n7 8 9 10 11 12 TI-Nspire™ Coding Student 50 min  \nIntroduction  \nThis activity is designed to build a series of beautiful mathematical visuals. The following mathematics is used to write the program:  \n• Modular Arithmetic  \n• Unit Circle (sine & cosine)  \n• Transformations  \nModular Arithmetic: A simple way to think of modular arithmetic is ‘remainder’. Try calculating mod(32,24) in the Calculator application.  \nIn the circle (opposite) draw lines to connect the ‘2 times’table.  \nExample: 1 connects to 2, 2 connects to 4,  \n3 connects to 6 …  \nA problem arises for products involving numbers greater than or equal to 12. Modular arithmetic can be used to fix this problem: mod(\\#,24) . Complete the mapping from 2 􀁵 0 through to 2 􀁵 23 to generate a ‘low resolution’ pattern. A program can be used to generate a much higher resolution of the pattern, and others in the family.  \nBuilding the Foundations   \nStart a new document and insert a basic program , call the program: StringArt  \nIf your program appears as a ‘split page’ press: Ctrl + 6 to ungroup the pages.  \n\n| The first step is to generate the 2 times table mapping. A set of numbers can be generated using a ‘loop’. The start and end values are both known, so a “FOR”loop is appropriate.\u003Cbr>Menu > Control > For … EndFor The “Disp” command can be used to display the numbers:\u003Cbr>Menu > I/O > Disp\u003Cbr>The ‘wait’command can be added to slow the program down:\u003Cbr>Menu > Hub > Wait Press Ctrl + R to run the program. | |\n| --- | --- |\n| The location of each point will be referenced using the unit circle.\u003Cbr>The 2 times tables values need to be converted to angles, the circle diagram includes these angles (degrees) .\u003Cbr>These values will be stored in variables:\u003Cbr>a1 (Angle for angle 1) & a2,(angle for angle 2)\u003Cbr>Edit the program to include these new variables and update the “Disp” command to show the angles.\u003Cbr>• The modular arithmetic is performed for 360 (degrees)\u003Cbr>• The multiplier: 15 comes from 360􀁱 􀁹 24 = 15􀁱 . | |\n| Transformations & the Unit Circle |  |\n| Unlike the cartesian plane, the drawing canvas on TI-nspire has the origin located in the top left corner. The canvas dimensions are 318 􀁵 212 pixels. To place the ‘string art’diagram in the centre of the screen, a simple set of transformations can be performed:\u003Cbr>h := 159 (Horizontal Transformation)\u003Cbr>v := 106 (Vertical Transformation) A dilation can scale the ‘unit’ circle into something more appropriate:\u003Cbr>r := 100 (Dilation)\u003Cbr>The transformations can be made at the start of the program. (See below right) | |\n| The angle can now be converted from degrees into radians.\u003Cbr>a1 :=15 i 􀁓 / 180 [Note that 􀜽2 is simply double 􀜽1] | |\n| Each line consists of a starting point: (􀝔1, 􀝕1) and end point: (􀝔2, 􀝕2), for simplicity, these can be done individually:\u003Cbr>􀝔1 ≔ ℎ + 􀝎􀜿􀝋􀝏 (􀜽1)\u003Cbr>􀝕1 ≔ 􀝒 + 􀝎􀝏􀝅􀝊 (􀜽1)\u003Cbr>􀝔2 ≔ ℎ + 􀝎􀜿􀝋􀝏 (􀜽2)\u003Cbr>􀝕2 ≔ 􀝒 + 􀝎􀝏􀝅􀝊 (􀜽2) | |\n\nVariables: x1, y1 , x2 andy2 are used in the program. The same variables are aligned to ‘parametric’equations in the Graphs application. These variables can be declared as “Local” at the start of the program, this means they only exist in the program. Alternatively, when graphing a parametric equation, start at: x3(t), y3(t)  \nThe last command is to draw the line on the canvas.  \nMenu > Draw > Shapes > DrawLine The syntax for the DrawLine command is logical:  \nDrawLine x1 , y1 , x2 , y2  \nPress Ctrl + R to run the program.  \n| Time to Explore |  |\n| --- | --- |\n| Now that your program has finished, try making a few simple changes:\u003Cbr>• Change the “For” loop to count up to 120. Change the corresponding a1 calculation then run the program. Explore different loop counts, with corresponding changes to the a1 value.\u003Cbr>• Explore what happens if the a2 multiplier is changed from‘2’ to ‘3’.\u003Cbr>• Just for fun: Change the line colour using the SetColor R, G, B command,(Red, Green, Blue) . The patterns get even better if the colour","cbCaio0vhmVPBFZo","https://ap.wps.com/l/cbCaio0vhmVPBFZo","pdf",746981,"English","# Introduction\n## Building the Foundations\n## Transformations & the Unit Circle\n## Time to Explore\n## Investigation\n## Questions & Answers","[{\"question\":\"How does modular arithmetic help generate the unit-circle string-art pattern?\",\"answer\":\"It uses remainder (mod) to map multiplication results into a fixed range, avoiding problems when products reach values at or above 12. This mapping can be generated directly for low resolution or via a program for higher resolution patterns.\"},{\"question\":\"What transformations are used to center and scale the diagram on TI-Nspire?\",\"answer\":\"Because the TI-Nspire drawing origin is at the top left, the notes apply a horizontal shift h := 159 and vertical shift v := 106 to center the figure, then use a dilation r := 100 to scale the unit circle appropriately.\"},{\"question\":\"What locus and parameter equations result when the outer circle traces around the inner circle with R = r?\",\"answer\":\"With point P at the outer-circle center, P traces a circle with x(t)=4cos(t) and y(t)=4sin(t) (CCW) or y(t)=-4sin(t) (CW). For a point on the outer circle, the path is x(t)=4cos(t)+2cos(2t) and y(t)=4sin(t)+2sin(2t).\"}]","String Art on the Unit Circle - Teacher Notes & Answers | PDF"]