{"id":184,"date":"2024-07-01T14:10:48","date_gmt":"2024-07-01T14:10:48","guid":{"rendered":"http:\/\/localhost\/teachingsecondarymathematics\/?post_type=content&#038;p=184"},"modified":"2026-08-26T12:59:25","modified_gmt":"2026-08-26T12:59:25","slug":"chapter-8-discovery","status":"publish","type":"content","link":"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/","title":{"rendered":"Chapter 8 &#8211; Discovery"},"content":{"rendered":"\n<div class=\"wp-block-group alignwide has-base-background-color has-background has-global-padding is-layout-constrained wp-container-core-group-is-layout-15050843 wp-block-group-is-layout-constrained\" style=\"margin-top:0;padding-top:var(--wp--preset--spacing--60);padding-right:var(--wp--preset--spacing--60);padding-bottom:var(--wp--preset--spacing--60);padding-left:var(--wp--preset--spacing--60)\">\n<div class=\"wp-block-group alignwide has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group alignwide has-global-padding is-content-justification-left is-layout-constrained wp-container-core-group-is-layout-54d22900 wp-block-group-is-layout-constrained\"><nav style=\"--separator: &quot;\/&quot;\" class=\"wp-block-breadcrumbs\" aria-label=\"Breadcrumbs\"><ol><li><a href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/\">Home<\/a><\/li><li><a href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/\">Resources<\/a><\/li><li><span aria-current=\"page\">Chapter 8 &#8211; Discovery<\/span><\/li><\/ol><\/nav><\/div>\n\n\n\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:60%\"><h1 class=\"alignwide wp-block-post-title has-large-font-size\">Chapter 8 &#8211; Discovery<\/h1>\n\n\n<p class=\"wp-block-paragraph\">Discovery learning is a method of indirect instruction. The teacher structures a learning environment that allows students to develop conclusions. Normally, when doing a direct instruction lesson, students know the teacher will eventually \u201ctell the big secret, formula, or conclusion.\u201d Students quickly learn there is no need to work at discovering. Teachers need to provide learning time, appropriate tools, and prompts and expect students to arrive at conclusions. This means teachers should not \u201ctell\u201d the answer.DOWNLOAD<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:40%\">\n<media-player  load=\"visible\" id=\"vidstack-video-block_5f7c423b66bfcad7f0bdcacbfd43f0c9\" class=\"vidstack-video\" title=\"Chapter 8 - Introduction\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-8\/Chapter8-Intro.mp4\">\n  <media-provider>\n                  <media-poster\n          class=\"vds-poster\"\n          alt=\"\">\n        <\/media-poster>\n\n  <\/media-provider>\n  <media-video-layout><\/media-video-layout>\n  <media-controls class=\"vds-controls\">\n      <media-controls-group class=\"vds-controls-group\"><\/media-controls-group>\n  <\/media-controls>\n<\/media-player><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group has-base-background-color has-background has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"margin-top:0;padding-top:var(--wp--preset--spacing--70);padding-bottom:var(--wp--preset--spacing--70)\">\n<div class=\"wp-block-group alignwide is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:66.66%\">\n<h2 id=\"aioseo-chapter-1-exercises\" class=\"wp-block-heading\">Exercises<\/h2>\n\n\n\n<div class=\"wp-block-group has-border-color has-black-border-color has-base-element-background-color has-background is-layout-flow wp-block-group-is-layout-flow\" style=\"border-width:2px;padding-top:var(--wp--preset--spacing--30);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--30);padding-left:var(--wp--preset--spacing--50)\">\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-1\" class=\"wp-block-heading\">Exercise 8.1<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Summarize and react to at least three major points raised in one of these publications: Everybody Counts, Curriculum and Evaluation Standards for School Mathematics, Professional Standards for Teaching Mathematics, Assessment Standards, Standards 2000, or Reshaping the Schools. Are the suggestions practical? Do you think they could have been instituted in your high school? Why or why not?<\/li>\n\n\n\n<li>Complete an annotated bibliography on all documents mentioned in part 1 of this exercise.<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-2\" class=\"wp-block-heading\">Exercise 8.2<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Use dynamic geometry software to construct a quadrilateral. Experiment by moving the vertices to show that the sum of the exterior angles will always be 360\u00b0.<\/li>\n\n\n\n<li>If the quadrilateral from part 1 is concave, what is the impact on the sum of the exterior angles? How do you rationalize this with students?<\/li>\n\n\n\n<li>Construct a regular 12-gon (dodecagon) using dynamic geometry software. Show that the sum of the measures of the exterior angles of this 12-gon is 360\u00b0. Describe the advantages and disadvantages of increasing the number of sides of the <em>n<\/em>-gon. What is the impact of this construction on determining for students that a circle is 360\u00b0 in full rotation?<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-3\" class=\"wp-block-heading\">Exercise 8.3<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Develop or find a number trick using at least four instructions (remember to give appropriate bibliographical credit). Have someone do the trick and record their reaction.<\/li>\n\n\n\n<li>Will complex numbers work for your trick? What conditions must be placed on your trick?<\/li>\n\n\n\n<li>Should calculators be allowed for students when a number trick is presented?&nbsp; Justify your reasoning. Ask two practicing teachers of mathematics the same question. Did their response surprise you?<\/li>\n\n\n\n<li>Create a number trick where the solution is the same for all participants regardless of the value each participant selects as a starting value. Try this with a group of students. What percentage of students obtained the correct solution? Did this surprise you? Why or why not?<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-4\" class=\"wp-block-heading\">Exercise 8.4<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Are there sets of values where commutativity of subtraction does exist?<\/li>\n\n\n\n<li>Does commutativity for multiplication exist in all number systems? Don\u2019t forget to consider the reals, complex, and quaternions.<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-5\" class=\"wp-block-heading\">Exercise 8.5<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Research howlers. Provide at least two more examples similar to.<\/li>\n\n\n\n<li>Generalize your discussion of howlers from part 1 of this exercise.<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-6\" class=\"wp-block-heading\">Exercise 8.6<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Another example similar to the development just done for \u201creduce\u201d can be made for the expression \u201cGive me a number larger than 1.\u201d What would be encountered as this question is investigated? How could the pitfalls be avoided, or extended?<\/li>\n<\/ol>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-exercise-8-7\" class=\"wp-block-heading\">Exercise 8.7<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Can twiddle be an arithmetic mean? (<em>B = D<\/em> in the original definition).<\/li>\n\n\n\n<li>Can twiddle be a harmonic mean? (<em>A = C<\/em> in the original definition).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n\n\n\n<h2 id=\"aioseo-problem-solving-challenges\" class=\"wp-block-heading\">Problem Solving Challenges<\/h2>\n\n\n\n<div class=\"wp-block-group has-border-color has-black-border-color has-base-element-background-color has-background is-layout-flow wp-block-group-is-layout-flow\" style=\"border-width:2px;padding-top:var(--wp--preset--spacing--30);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--30);padding-left:var(--wp--preset--spacing--50)\">\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"wp-block-paragraph\">1.&nbsp; A camel merchant willed his 17 camels to his three sons. In the merchant\u2019s will, the camels were to be divided among them as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The eldest son was to receive half of the camels.<\/li>\n\n\n\n<li>The middle son was to receive a third of the camels.<\/li>\n\n\n\n<li>The youngest son was to receive a ninth of the camels.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The executor of the merchant\u2019s estate was perplexed. Finally, he devised a method for dividing the camels without having to slaughter any of the animals. How many camels did each son receive? Explain his solution. Hint: What would be the result if you had 18 camels?<\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>Answer<\/summary>\n<p class=\"wp-block-paragraph\">Answer\/solution:&nbsp; The eldest son received 9 camels, the middle son received 6, and the youngest received 2. The executor added one camel to the total giving 18 camels, which can be divided by 2, 3, and 9. Once the will was executed, the camel that had been added was still the property of the executor and not a part of the will.<\/p>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"wp-block-paragraph\">2.&nbsp; Suppose the NCAA decided to have a single elimination tournament involving all Division 1A teams at the end of the basketball season. If there are 303 Division 1A teams, how many games will be played before a national champion is declared?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hint:&nbsp; Try a smaller problem with 6 teams.&nbsp; 10 teams.<\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>Answer<\/summary>\n<p class=\"wp-block-paragraph\">Answer\/solution: 302 games to declare a winner. Since this is a single elimination tournament, each loss sends some team home. Out of the 303 teams, there is only one national champion, which means there must be 302 losses or 302 games to declare the winner. You could say the following:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">302 teams means 151 games (one team has a bye)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">152 teams means 76 games (bye team back in)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">76 teams means 38 games<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">38 teams means 19 games<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">18 teams means 9 games (one team has a bye)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">10 teams means 5 games (bye team back in)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">4 teams means 2 games (one team has a bye)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2 teams means 1 game (one team has a bye)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2 teams means 1 game (bye team back in)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">TOTAL&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 302<\/p>\n<\/details>\n<\/div>\n<\/div>\n\n\n\n<h2 id=\"aioseo-additional-learning-activities\" class=\"wp-block-heading\">Additional Learning Activities<\/h2>\n\n\n\n<div class=\"wp-block-group has-border-color has-black-border-color has-base-element-background-color has-background is-layout-flow wp-block-group-is-layout-flow\" style=\"border-width:2px;padding-top:var(--wp--preset--spacing--50);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--50);padding-left:var(--wp--preset--spacing--50)\">\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"wp-block-paragraph\">There is no Additional Learning Activities for this chapter.<\/p>\n<\/div>\n<\/div>\n\n\n\n<h2 id=\"aioseo-videos\" class=\"wp-block-heading\">Videos<\/h2>\n\n\n\n<div class=\"wp-block-group has-border-color has-black-border-color has-base-element-background-color has-background is-layout-flow wp-block-group-is-layout-flow\" style=\"border-width:2px;padding-top:var(--wp--preset--spacing--50);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--50);padding-left:var(--wp--preset--spacing--50)\">\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-columns alignwide are-vertically-aligned-top is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-group is-vertical is-layout-flex wp-container-core-group-is-layout-2c90304e wp-block-group-is-layout-flex\">\n<media-player  load=\"visible\" id=\"vidstack-video-block_9b44f35afb1cc9a38734226e9e7a121a\" class=\"vidstack-video\" title=\"Problem Solving\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-8\/Chapter8-PSC.mp4\">\n  <media-provider>\n                  <media-poster\n          class=\"vds-poster\"\n          alt=\"\">\n        <\/media-poster>\n\n  <\/media-provider>\n  <media-video-layout><\/media-video-layout>\n  <media-controls class=\"vds-controls\">\n      <media-controls-group class=\"vds-controls-group\"><\/media-controls-group>\n  <\/media-controls>\n<\/media-player>\n\n\n<h3 id=\"aioseo-problem-solving\" class=\"wp-block-heading\">Problem Solving<\/h3>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-group is-vertical is-layout-flex wp-container-core-group-is-layout-fba139f5 wp-block-group-is-layout-flex\">\n<media-player  load=\"visible\" id=\"vidstack-video-block_8c0ee531ab4072f83504d30459780a21\" class=\"vidstack-video\" title=\"Sticky Question\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-8\/Chapter8-Sticky.mp4\">\n  <media-provider>\n                  <media-poster\n          class=\"vds-poster\"\n          alt=\"\">\n        <\/media-poster>\n\n  <\/media-provider>\n  <media-video-layout><\/media-video-layout>\n  <media-controls class=\"vds-controls\">\n      <media-controls-group class=\"vds-controls-group\"><\/media-controls-group>\n  <\/media-controls>\n<\/media-player>\n\n\n<h3 id=\"aioseo-sticky-question\" class=\"wp-block-heading\">Sticky Question<\/h3>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:33.33%\">\n<p class=\"wp-block-paragraph\"><strong>On this page<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-base-background-color has-background is-horizontal is-content-justification-space-between is-nowrap is-layout-flex wp-container-core-group-is-layout-9c6c4bab wp-block-group-is-layout-flex wp-container-1 is-position-sticky\">\n<nav class=\"wp-block-table-of-contents\"><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-chapter-1-exercises\">Exercises<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-1\">Exercise 8.1<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-2\">Exercise 8.2<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-3\">Exercise 8.3<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-4\">Exercise 8.4<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-5\">Exercise 8.5<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-6\">Exercise 8.6<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-exercise-8-7\">Exercise 8.7<\/a><\/li><\/ol><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-problem-solving-challenges\">Problem Solving Challenges<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-additional-learning-activities\">Additional Learning Activities<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-videos\">Videos<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-problem-solving\">Problem Solving<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/#aioseo-sticky-question\">Sticky Question<\/a><\/li><\/ol><\/li><\/ol><\/nav>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Discovery learning is a method of indirect instruction. The teacher structures a learning environment that allows students to develop conclusions. Normally, when doing a direct instruction lesson, students know the teacher will eventually \u201ctell the big secret, formula, or conclusion.\u201d Students quickly learn there is no need to work at discovering. Teachers need to provide [&hellip;]<\/p>\n","protected":false},"featured_media":0,"parent":112,"menu_order":0,"template":"no-title-2","format":"standard","meta":{"_acf_changed":false},"categories":[],"class_list":["post-184","content","type-content","status-publish","format-standard","hentry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Discovery learning is a method of indirect instruction. The teacher structures a learning environment that allows students to develop conclusions. Normally, when doing a direct instruction lesson, students know the teacher will eventually \u201ctell the big secret, formula, or conclusion.\u201d Students quickly learn there is no need to work at discovering. Teachers need to provide\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-8-discovery\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO Pro (AIOSEO) 5.0.0.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"Teaching Secondary Mathematics - 5th Edition\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Chapter 8 \u2013 Discovery - Teaching Secondary Mathematics\" \/>\n\t\t<meta property=\"og:description\" content=\"Discovery learning is a method of indirect instruction. 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