{"id":69,"date":"2024-05-24T13:22:24","date_gmt":"2024-05-24T13:22:24","guid":{"rendered":"http:\/\/localhost\/teachingsecondarymathematics\/?post_type=content&#038;p=69"},"modified":"2026-08-26T13:01:53","modified_gmt":"2026-08-26T13:01:53","slug":"chapter-2-learning-theory-curriculum-and-assessment","status":"publish","type":"content","link":"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/","title":{"rendered":"Chapter 2 &#8211; Learning Theory, Curriculum, and Assessment"},"content":{"rendered":"\n<div class=\"wp-block-group 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 2 &#8211; Learning Theory, Curriculum, and Assessment<\/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 2 &#8211; Learning Theory, Curriculum, and Assessment<\/h1>\n\n\n<p class=\"wp-block-paragraph\">Ed Begle said, \u201cWe have learned a lot about teaching better mathematics but not much about teaching mathematics better.\u201d When perusing the history of teaching mathematics, it appears as if we constantly look for a magic method or strategy that will serve all students. There is no enchanted formula that fits every learner. We need to realize that a variety of methods and strategies are necessary to meet the mathematical learning needs of all students. Our task, as teachers of mathematics, is to determine which method will be most beneficial to the mathematical learning of each student and when these strategies will be most effective. If a teacher uses a strategy with no successful result from the learner, at what point is that method abandoned in favor of another? The decision is most often influenced by our own background, training, experiences, bias, and current curriculum.<\/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_1435fb954f531e1ace7cf285dd48db88\" class=\"vidstack-video\" title=\"Chapter 2 - Introduction\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-2\/Chapter-2--Introduction.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-2-1\" class=\"wp-block-heading\">Exercise 2.1<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>What are the major characteristics you would ascribe to a positive mathematics classroom? Which of these would you control? Which would be dependent on your students? Which of these would be dependent on administration?<\/li>\n\n\n\n<li>Was the mathematics learning environment you experienced in your secondary school years constructivist or behaviorist based? Describe your experiences to amplify your selection.<\/li>\n\n\n\n<li>List three similarities and three differences found in the behaviorist and constructivist mathematics classroom.<\/li>\n\n\n\n<li>Constructivism and behaviorism are idealistic. Can either of them be effective in the real world of secondary schools? Why or why not? Should there be a blending of the two?<\/li>\n\n\n\n<li>Describe your mathematics classroom of the future.<\/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-2-2\" class=\"wp-block-heading\">Exercise 2.2<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Select an activity from one of the books in the Addenda Series. Critique the activity. Will it appeal to students? Why or why not? Would you change it before using it? Why or why not? Are the readiness expectations necessary for the activity clear? Will the students be able to connect the activity to something in their world? Why or why not?<\/li>\n\n\n\n<li>List the titles of the books in the Addenda Series. Just using the list, which one sounds the most appealing to you and why?<\/li>\n\n\n\n<li>Select any book from the Addenda Series. Select any chapter description and summarize it. Be sure to include the commentaries listed as \u201cAssessment Matters,\u201d \u201cTeaching Matters,\u201d \u201cTry This,\u201d and so on for the whole chapter.<\/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-2-3\" class=\"wp-block-heading\">Exercise 2.3<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The Professional Standards (pp. 135\u2013139) list courses that should be a part of the background for teachers of mathematics in Grades pre-K\u201312. Focus on the courses for Grades 5\u20138 or Grades 9\u201312. Compare the coursework in your background with those listed. Elaborate on any differences and describe whether or not you feel they are significant.<\/li>\n\n\n\n<li>Select a vignette from the Professional Standards that you believe to be a description of a good classroom situation. Highlight the strong points of the vignette and describe your impression of the strengths.<\/li>\n\n\n\n<li>Select a vignette from the Professional Standards that you believe to be a description of a poor classroom situation. Highlight the weak points of the vignette and describe your impression of the weaknesses along with what could be done to strengthen them.<\/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-2-4\" class=\"wp-block-heading\">Exercise 2.4<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Read the \u201cWhat\u2019s Next?\u201d (pp. 81\u201383) section of the Assessment Standards. What do you think? Are the descriptions accurate? Were there items that should be added to or deleted from the discussion? Do you think you could build this type of assessment program in your classes and school when you begin teaching? Why or why not?<\/li>\n\n\n\n<li>Summarize each of the six standards (pp. 11, 13, 15, 17, 19, and 21 in the Assessment Standards). Indicate any section you feel is particularly useful or of little benefit, stating why you feel as you do.<\/li>\n\n\n\n<li>React to the highlighted description of the process used by Ms. Lee, Mr. Jackson, and Ms. Romario as they made decisions as described on pp. 36\u201339 of the Assessment Standards. If you had been in the situation, would you have made other suggestions or rejected any they made? Why or why not?<\/li>\n\n\n\n<li>The Assessment Standards contain a list of suggested major shifts from some things to others as a part of an effective assessment practice (p. 83). What do you think? Are there items that should have been added to or deleted from the discussion? Do you think you could build this type of assessment program in your classes and school when you begin teaching? 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-2-5\" class=\"wp-block-heading\">Exercise 2.5<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Compare and contrast the problem-solving and communication standards for Grades 9\u201312 in the Standards and Standards 2000. Is there a significant difference in the theme behind these standards?<\/li>\n\n\n\n<li>Review focal points for Grade 6, 7, or 8.&nbsp; Describe how the focal points enhance that particular grade for Standards 2000.<\/li>\n\n\n\n<li>Look at the standards for Grades 5\u20138 in the Standards and Standards 2000. Rank order the standards in each document as to their importance in the mathematics curriculum. Is there a significant difference in the order in which NCTM proposed them? Does it matter?<\/li>\n\n\n\n<li>Look at the standards for Grades 9\u201312 in the Standards and Standards 2000. Rank order the standards in each document as to their importance in the mathematics curriculum. Is there a significant difference in the order in which NCTM proposed them? Does it matter?<\/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-2-6\" class=\"wp-block-heading\">Exercise 2.6<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>How do the Common Core Standards differ from Standards 2000 for Grades 6, 7, and 8?<\/li>\n\n\n\n<li>How do the Common Core Standards differ from Standards 2000 for Grade 3?&nbsp; Explain the significant difference you discover.<\/li>\n\n\n\n<li>How do the Common Core Standards differ from Standards 2000 for Algebra?<\/li>\n\n\n\n<li>How do the Common Core Standards differ from Standards 2000 for Geometry?<\/li>\n\n\n\n<li>How do the Common Core Standards differ from Standards 2000 for Statistics and Probability?<\/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-2-7\" class=\"wp-block-heading\">Exercise 2.7<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Examine the problem sections of a secondary mathematics textbook. How many of the problems come from the world viewed by a student? Of all the problems, how many appear to be designed to appeal to girls?<\/li>\n\n\n\n<li>How much paint does it take to paint all the segments on a football field? Is this a good problem-solving problem? Why or why not? Does this problem reflect equity bias? If yes, what could be done to eliminate it? If no, 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-exercises-2-8\" class=\"wp-block-heading\">Exercise 2.8<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Is it reasonable to have a test of major proportions (like the national Algebra I test that was discussed) at some point during the school year? Why or why not? Describe the impact on the curriculum.<\/li>\n\n\n\n<li>Does your state have a <em>mandatory<\/em> mathematics test as a graduation requirement? If so, what is the content level of the test? Is the test based upon the highest mathematics course required by the state for graduation?<\/li>\n\n\n\n<li>If your state has required Algebra I test for graduation, is there a correlation between student scores on the test and grades given in the course? Should there be a strong correlation? Does it matter?&nbsp;<\/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-2-9\" class=\"wp-block-heading\">Exercise 2.9<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Is a national curriculum or text-driven curriculum desirable? Why or why not?<\/li>\n\n\n\n<li>If the curriculum framework requires a specific concept taught that is not addressed in the textbook assigned to the student, will you teach the concept?&nbsp; How will you supplement the text?&nbsp;<\/li>\n\n\n\n<li>Examine several textbooks for a given mathematics concept. Describe their similarities. Are there any significant differences? Is there a text that is notably different from the rest? If there is a different text, rationalize why it should or should not be available for adoption. If there is no different text, discuss why they are all the same.<\/li>\n\n\n\n<li>Are the mathematics concepts taught in high school driven by the state curriculum frameworks or the textbook used in the classroom? Justify your response.&nbsp;<\/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-2-10\" class=\"wp-block-heading\">Exercise 2.10<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Should you, the teacher, as the local authority on your class, solely determine the material to be covered in a given class? If yes, why? If no, how much outside influence should be acceptable and why?<\/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-2-11\" class=\"wp-block-heading\">Exercise 2.11<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>If you have been to a local, regional, or state mathematics contest, describe the emotional and general atmosphere of the setting. If you have not seen a competition, visit one as an observer and then describe what you noticed.<\/li>\n\n\n\n<li>Locate a secondary student who is currently a member of a school mathematics team or who has participated in a mathematics competition in the past year. Describe that student\u2019s reactions and feelings about the associated experiences.<\/li>\n\n\n\n<li>Locate a teacher who is sponsoring or has sponsored a mathematics competition team. Ask why they do it and what is in it for them.<\/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-2-12\" class=\"wp-block-heading\">Exercise 2.12<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Determine the error pattern the student made in each of the following problems:<\/li>\n<\/ol>\n\n\n\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mtable columnalign=\"right right right right\" rowspacing=\"4pt\" columnspacing=\"1em\" rowlines=\"none solid\">    <mtr>      <mtd>        <mn>4567<\/mn>      <\/mtd>      <mtd>        <mn>389<\/mn>      <\/mtd>      <mtd>        <mn>2468<\/mn>      <\/mtd>      <mtd>        <mn>3421<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mo>+<\/mo>        <mn>7968<\/mn>      <\/mtd>      <mtd>        <mo>+<\/mo>        <mn>964<\/mn>      <\/mtd>      <mtd>        <mo>+<\/mo>        <mn>3517<\/mn>      <\/mtd>      <mtd>        <mo>+<\/mo>        <mn>2476<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mn>14635<\/mn>      <\/mtd>      <mtd>        <mn>1453<\/mn>      <\/mtd>      <mtd>        <mn>7085<\/mn>      <\/mtd>      <mtd>        <mn>5897<\/mn>      <\/mtd>    <\/mtr>  <\/mtable><\/math>\n\n\n\n<p class=\"wp-block-paragraph\">Describe the error the student is making. List the steps you would employ to assist the student in learning how to do the problem correctly and avoid repeating the same error. Could this error have been caused because the student is not accustomed to seeing addition problems written horizontally?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Determine the error pattern the student made in each of the following problems:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;&nbsp; 4567 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;389&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;2468&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;3421<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">+7968&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; +964&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; +3517&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; +2476<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">14635&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1453&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;7085&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;5897<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Determine the error pattern the student made in each of the following problems:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">46.325 + 234.56 + 13.567 + 2.7964 = 0.000000111312<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">3.579 + 54.32 + 684.2 = 0.015853<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">35.234 + 67.531 = 0.102765<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">4.8 + 32 + 0.79 + 7.8 = 0.0237<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Describe the error the student is making. List the steps you would employ to assist the student in learning how to do the problem correctly and avoid repeating the same error.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Define an error pattern you think a student would make. State the grade level. Provide sufficient examples. Give your error pattern to a peer to solve. Describe your discussions with your peer about the error pattern, and how to correct it.<\/li>\n\n\n\n<li>Develop a continuum of concepts necessary for a student to be able to solve a problem like . Assume the students are able to work successfully with the four arithmetic operations.<\/li>\n<\/ul>\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--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\" style=\"padding-top:0;padding-bottom:0\">\n<h3 id=\"aioseo-1-pyramid-of-numbers\" class=\"wp-block-heading\">1. Pyramid of Numbers<\/h3>\n\n\n\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mtable rowspacing=\"4pt\" columnspacing=\"1em\">    <mtr>      <mtd>        <mn>1<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mn>11<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mn>21<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mn>1211<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mn>111221<\/mn>      <\/mtd>    <\/mtr>    <mtr>      <mtd>        <mn>312211<\/mn>      <\/mtd>    <\/mtr>  <\/mtable><\/math>\n\n\n\n<p class=\"wp-block-paragraph\">What are the next two lines?<\/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: 13112221 and 1113213211.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The next line is determined by reading the preceding one. Look at the top line \u201c1.\u201d Reading it, you could say \u201cThere is one one.\u201d Looking at the second line, you see \u201cone one.\u201d Looking at the second line you would say \u201cThere are two ones.\u201d Read that as \u201ctwo ones\u201d and you have the third line. Reading the third line, you would say \u201cThere is one two and one one\u201d which describes the numbers on line four. The fifth line is determined by reading the fourth line \u201cone one, one two, two ones\u201d or 111221. The fifth line indicates that numbers that are read as digits appear and are grouped by the same digits in that order. The sixth line comes from reading the fifth as \u201cThree ones followed by two twos which are followed by one one\u201d or 312211. Thus, the seventh line would be 13112221, and the eighth line would be 1113213211.<\/p>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<h3 id=\"aioseo-2-the-letter-mania\" class=\"wp-block-heading\">2. The Letter Mania<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Based on the following two groups of letters, place each letter from the following letters in its appropriate group: I, J, K, L:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Group 1: A, E, F, H<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Group 2: B, C, D, G<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">.<\/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: Group 1 letters are all straight segments while Group 2 contain curves. So, I, K, and L go into Group 1 while J goes into Group 2.<\/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\"><strong>1. Why are manhole covers round?<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Extension<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The text discussions and exercises should help you understand why manhole covers are round. They will not fall down the hole (assuming the diameter of the lid is slightly larger than the diameter of the hole). The circle is not the only shape that can serve as a manhole cover. A Reuleaux triangle in Figure 2.1 will work, too.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Construct an equilateral triangle, ABC. Using A as the center and B as the radius end, construct a minor arc between B and C. Place a point, E, on the arc, making segment AE. Repeat this construction for the remaining two sides of the original triangle to create segments BG and CJ.&nbsp; Segments AE, BG, and CJ are all the same length since they are radii of congruent circles. Thus no lid would fall down this hole.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-2\/Fig2-1.jpg\" alt=\"Figure 2.1\"\/><figcaption class=\"wp-element-caption\">Figure 2.1<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">This Reuleaux triangle water main cover was spotted on one of the back streets of St. Augustine, FL in July 2004. The historic district of St. Augustine is many square blocks. Check a few blocks off of the main street in front of the Methodist church, for sure. There are several others throughout the historic district.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-2\/Fig2-2.jpg\" alt=\"Figure 2.2\"\/><figcaption class=\"wp-element-caption\">Figure 2.2<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Could a Reuleaux square serve as a manhole? Construct a square. Select two adjacent vertices and make a circle. The arc is not outside the square so this cannot be used to create a Reuleaux square. Select one side and construct its midpoint. Use the midpoint as the center of a circle whose radius length is determined by one of the vertices not on the midpoint\u2019s segment. That circle passes through two vertices.&nbsp; But the radius of the circle will be <math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mfrac>\n    <mrow>\n      <mi>s<\/mi>\n      <msqrt>\n        <mn>3<\/mn>\n      <\/msqrt>\n    <\/mrow>\n    <mn>2<\/mn>\n  <\/mfrac>\n<\/math> while the diagonal length is <math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <mi>s<\/mi>\n    <msqrt>\n      <mn>2<\/mn>\n    <\/msqrt>\n  <\/mrow>\n<\/math>. The diagonal is longer and the lid could fall down the hole. So, this construction in Figure 2.3 cannot be used to create a Reuleaux square.&nbsp;&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-2\/Fig2-3.jpg\" alt=\"Figure 2.3\"\/><figcaption class=\"wp-element-caption\">Figure 2.3<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Could a Reuleaux regular pentagon serve as a manhole? Create a regular pentagon. Join a vertex with a vertex from a non-adjacent side and construct the short arc between two adjacent vertices, putting a point on the arc. Repeat this for the other four sides as shown in Figure 2.4. The five segments are all the same length since they are radii of congruent circles. Thus no lid would fall down this hole.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-2\/Fig2-4.jpg\" alt=\"Figure 2.4\"\/><figcaption class=\"wp-element-caption\">Figure 2.4<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Your Turn<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Create a Reuleaux figure from a regular heptagon as was done with the regular pentagon.&nbsp; What conclusions can be made about Reuleaux figures generated from regular n-gons when n is odd?<\/li>\n<\/ol>\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-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:50%\">\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_e5f171fc81185dc9e120f34ec1c7b623\" 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-2\/Chapter-2--Problem-Solving.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-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\">\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_6ae9e49edbee862cab2c723b974c041c\" 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-2\/Chapter-2--Sticky-Question.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\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\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-chapter-1-exercises\">Exercises<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-1\">Exercise 2.1<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-2\">Exercise 2.2<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-3\">Exercise 2.3<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-4\">Exercise 2.4<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-5\">Exercise 2.5<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-6\">Exercise 2.6<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-7\">Exercise 2.7<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercises-2-8\">Exercise 2.8<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-9\">Exercise 2.9<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-10\">Exercise 2.10<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-11\">Exercise 2.11<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-exercise-2-12\">Exercise 2.12<\/a><\/li><\/ol><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-problem-solving-challenges\">Problem Solving Challenges<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-1-pyramid-of-numbers\">1. Pyramid of Numbers<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-2-the-letter-mania\">2. The Letter Mania<\/a><\/li><\/ol><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-additional-learning-activities\">Additional Learning Activities<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-videos\">Videos<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#aioseo-problem-solving\">Problem Solving<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-2-learning-theory-curriculum-and-assessment\/#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>Ed Begle said, \u201cWe have learned a lot about teaching better mathematics but not much about teaching mathematics better.\u201d When perusing the history of teaching mathematics, it appears as if we constantly look for a magic method or strategy that will serve all students. There is no enchanted formula that fits every learner. We need [&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-69","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=\"Ed Begle said, \u201cWe have learned a lot about teaching better mathematics but not much about teaching mathematics better.\u201d When perusing the history of teaching mathematics, it appears as if we constantly look for a magic method or strategy that will serve all students. There is no enchanted formula that fits every learner. 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