[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-508151-105":59,"doc-detail-508151-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","pert-and-nert-lesson-2-pert-and-nert","Pert and Nert - Lesson 2: Pert and Nert","","Lesson 2, titled “Pert and Nert,” guides students through exponential growth and decay using matching tasks between exponential equations and graphs, sorting by behavior, and analyzing key characteristics. Students work with exponential and logarithmic functions, including function attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and extrema. The lesson derives the natural base e via the compound interest formula, then applies e to model continuous compounding for interest and population growth, including prediction tasks and comparisons of continuous versus discrete compounding.",{"@graph":69,"@context":123},[70,84,106],{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/pert-and-nert-lesson-2-pert-and-nert/508151/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/pert-and-nert-lesson-2-pert-and-nert/508151.png","ImageObject",300,407,{"name":92,"@type":93},"Muffet","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-10-05",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do students determine whether an exponential graph represents growth or decay?","Question",{"text":113,"@type":114},"Students match graphs to equations, then sort each function based on whether its behavior shows exponential growth or exponential decay. They use graph direction and characteristics tied to the b-value to justify the classification.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the constant e connected to compound interest in this lesson?",{"text":118,"@type":114},"Students use the compound interest expression with increasing compounding periods to recognize that the recurring factor approaches the irrational number e. This leads to the continuous compounding form A = Pert.",{"name":120,"@type":111,"acceptedAnswer":121},"How does the lesson apply e to model population change?",{"text":122,"@type":114},"Students transfer the continuous-compounding idea to the population model N(t) = N0ert. They explain how the components support growth or decline and then use the resulting function to make predictions.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},508151,1791439459,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},962090768927,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","2  \nPert and Nert  \nProperties of Exponential Graphs  \nMATERIALS  \nGluesticks Graphing technology Scissors  \nLesson Overview  \nStudents match exponential equations and their graphs, sort them by whether they are increasing or decreasing, and compare the key characteristics of increasing and decreasing exponential functions. They use the compound interest formula to derive the value of the constant e. Students then apply the constant e in the formula for interest compounded continuously and population growth.  \nAlgebra 2  \nAttributes of Functions and Their Inverses  \n(2) The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:  \n(A) graph the functions f(x) 5 √__x, f (x) 5 , f (x) 5 x3, f (x) 5 3√__x, f (x) 5 bx, f (x) 5 | x | , and  \nf (x) 5 logb (x), where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.  \nExponential and Logarithmic Functions and Equations  \n(5) The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:  \n(B) formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation.  \nELPS  \n1.A, 1. B, 1. D, 1. E, 1.G, 1. H, 2.C, 2. D, 2. F, 2.G, 2. H, 2. I, 3.A, 3. B, 3.C, 3. D, 3. E, 3. F, 3. H, 3.J, 4.A, 4. B, 4.C, 4.G, 4. H, 4. I, 4. K, 5. D, 5. E  \nEssential Ideas  \n• For basic exponential growth functions, f(x) 5 bx, b is a value greater than 1. For basic exponential decay functions, f(x) 5 bx, b is a value between 0 and 1.  \nLESSON 2: Pert and Nert • 767A  \n• The compound interest formula is A 5 P ? ( 1 1 )kt, where A represents the value, P represents the principal amount, r represents the interest rate, and k represents the frequency of compounding in time t.  \n• The natural base e ≈ [2.7182818 . . . is](2.7182818 . . . is) an irrational number, also known as Euler's number.  \n• The formula for compound interest with continuous compounding is A 5 Pert, where P represents the principal, r represents the interest rate, and t represents time in years.  \n• The formula for population growth is N(t) 5 N0 ert, where N0 represents the initial population, r represents the rate of growth, and t represents time in years.  \n767B • TOPIC 1: Exponential and Logarithmic Functions  \nLesson Structure and Pacing: 2 Days  \nDay 1  \nEngage  \nGetting Started: You Have the Power  \nStudents match exponential graphs and their equations. They then sort them according to whether they model exponential growth or decay and complete a table for each function.  \nStudents also address similarities and differences among exponential growth and decay functions.  \nDevelop  \nActivity 2.1: Key Characteristics of Exponential Functions  \nStudents analyze whether the b-value of an exponential function models growth or decay. They write exponential functions that satisfy given characteristics. Students then summarize and compare the key characteristics of exponential functions that model growth and decay.  \nDay 2  \nActivity 2.2: The Natural Base e  \nStudents use the compound interest formula A 5 P ? (1 + )kt to demonstrate how the value changes as the number of compounding periods per year increases. They use specific parameters to recognize that the expression ()k approaches the irrational number, e, as k gets increasingly large. This is used to derive the formula for continuously compounding interest, A 5 Pert. Students also approximate the graph off(x) 5 ex, approximate a value from the graph, and they use the formula A 5 Pert to solve problems.  \nActivity 2.3: Modeling Population Change With e  \nStudents transfer their understanding of the formula for interest compounded continuou","cbCaikH46XovoFkZ","https://ap.wps.com/l/cbCaikH46XovoFkZ","pdf",1628572,28,"English","# Lesson Overview\n## Students’ core tasks\n# Essential Ideas\n## Growth and decay models\n## Deriving and applying e\n# Lesson Structure and Pacing\n## Day 1: Engage and Develop\n## Day 2: Activities 2.1–2.3\n# Demonstrate\n## Talk the Talk: Easy E\n# Engage Facilitation Notes\n## Activity goals and questions","[{\"question\":\"How do students determine whether an exponential graph represents growth or decay?\",\"answer\":\"Students match graphs to equations, then sort each function based on whether its behavior shows exponential growth or exponential decay. They use graph direction and characteristics tied to the b-value to justify the classification.\"},{\"question\":\"How is the constant e connected to compound interest in this lesson?\",\"answer\":\"Students use the compound interest expression with increasing compounding periods to recognize that the recurring factor approaches the irrational number e. This leads to the continuous compounding form A = Pert.\"},{\"question\":\"How does the lesson apply e to model population change?\",\"answer\":\"Students transfer the continuous-compounding idea to the population model N(t) = N0ert. They explain how the components support growth or decline and then use the resulting function to make predictions.\"}]","Pert and Nert - Lesson 2: Pert and Nert | PDF",1791210409,71]