{"id":171,"date":"2024-07-01T13:43:30","date_gmt":"2024-07-01T13:43:30","guid":{"rendered":"http:\/\/localhost\/teachingsecondarymathematics\/?post_type=content&#038;p=171"},"modified":"2026-08-26T12:58:24","modified_gmt":"2026-08-26T12:58:24","slug":"chapter-5-skills-in-teaching-mathematics","status":"publish","type":"content","link":"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/","title":{"rendered":"Chapter 5 &#8211; Skills in Teaching Mathematics"},"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 5 &#8211; Skills in Teaching Mathematics<\/span><\/li><\/ol><\/nav><\/div>\n\n\n\n<div class=\"wp-block-columns alignwide has-small-font-size 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 5 &#8211; Skills in Teaching Mathematics<\/h1>\n\n\n<p class=\"wp-block-paragraph\">How wonderful it would be if we knew everything about students, teaching, mathematics, and the teaching of mathematics. We could bottle it, sell it, become rich, and solve a lot of problems for everyone in the process. We all know there is no magic formula for teaching mathematics. In the process of attempting to find keys, we often stumble and miss objectives, but we also learn a variety of things that can be cataloged and used at a later date in other settings. This chapter deals with a broad collection of ideas, concepts, and strategies that should be useful in a variety of teaching environments. As you work through your career and attempt these suggestions, you will compile a list of methods and activities that will help your students learn mathematics.<\/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<div class=\"wp-block-group has-border-color has-black-border-color has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-width:2px\">\n<media-player  load=\"visible\" id=\"vidstack-video-block_33956af4cec3fc17da84be07c046d8a6\" class=\"vidstack-video\" title=\"Chapter 5 - Introduction\" src=\"https:\/\/s3-eu-west-1.amazonaws.com\/s3-euw1-ap-pe-ws4-cws-documents.ri-prod\/9781032028439\/chapter-5\/Chapter5-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<\/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&nbsp;<\/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-5-1\" class=\"wp-block-heading\">Exercise 5.1<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Describe how statistics and algebra could be related in a middle school setting.<\/li>\n\n\n\n<li>Relate number theory and geometry in an environment appropriate for a geometry student.<\/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-5-2\" class=\"wp-block-heading\">Exercise 5.2<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Find a number trick that could be used to motivate students to learn the standard addition algorithm.<\/li>\n\n\n\n<li>Find a number trick that could be used to motivate students to learn the standard subtraction algorithm.<\/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-5-3\" class=\"wp-block-heading\">Exercise 5.3<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>List five weird questions that could be used to stimulate students to learn mathematics. At least two should be appropriate for middle school and at least two for high school.<\/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-5-4\" class=\"wp-block-heading\">Exercise 5.4<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Do the tower puzzle for five disks and show the solution using base 2 numeration.<\/li>\n\n\n\n<li>Can you define a process that will predict where to move a disk? That is, if the three pegs the disks are placed on are named A, B, and C, devise a system that would say where a specific disk in the sequence of moves would be placed.<\/li>\n\n\n\n<li>There is a similar rule for the base 2 numeration solution of the tower puzzle using 1s rather than 0s. Describe it.<\/li>\n\n\n\n<li>Locate two additional websites that provide interactive versions of the Tower of Hanoi.&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-5-5\" class=\"wp-block-heading\">Exercise 5.5<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Define the generalized formula for the <em>n<\/em><sup>th<\/sup> heptagonal and octagonal numbers.<\/li>\n\n\n\n<li>Describe connections for pentagonal or hexagonal number sets as was done for triangular numbers, the handshake problem, and joining vertices of a convex <em>n<\/em>-gon with line segments.<\/li>\n\n\n\n<li>Define a generalization for the common differences of each counting number column in Figure 5.5 of the text.<\/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-5-6\" class=\"wp-block-heading\">Exercise 5.6<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Why do we teach the sequence Algebra I, Geometry, Algebra II?<\/li>\n\n\n\n<li>What are the names of the women who invented the bulletproof vest, fire escapes, windshield wipers, and laser printers?<\/li>\n\n\n\n<li>Identify another set of numbers that will react like 13 \u00d7 7 = 28 in a similar setting.<\/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-5-7\" class=\"wp-block-heading\">Exercise 5.7<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Do the problem 16,873 \u00f7 47 using long division. List all the potential problem areas or places where a student might be expected to make errors.<\/li>\n\n\n\n<li>Ask a group of students to write a personal answer to the following question, \u201cWhat do you want to learn?\u201d&nbsp; Allow 5{en}10 minutes to write an answer.&nbsp; Do not put restrictions on the question such as \u201c\u2026in math class, what do you want to learn?\u201d Review the student responses. What did you learn?<\/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-5-8\" class=\"wp-block-heading\">Exercise 5.8<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Provide a paragraph-style summary of the last meeting of the class you are using this text with. Swap papers with another student from the class. How similar are the papers? What, if any, are the differences?<\/li>\n\n\n\n<li>Locate a website other than MacTutor that can be used as an excellent resource or reference for the history of mathematics.<\/li>\n\n\n\n<li>Create a word problem that would appeal to a student in a secondary mathematics setting.<\/li>\n\n\n\n<li>Pick a secondary textbook explanation you feel is unclear. Rewrite it, eliminating all areas of concern. Compare your explanation with that of the author\u2019s and describe the strengths and weaknesses of each.<\/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-5-9\" class=\"wp-block-heading\">Exercise 5.9<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Describe some of the earliest uses of numbers and numerals.<\/li>\n\n\n\n<li>Document the beginning of irrational numbers and how they were used in early computations.<\/li>\n\n\n\n<li>Research at least one of the following topics and prepare a written summary of its development: Egyptian pyramids, golden section, golden ratio, Fibonacci numbers, networks, twisted surfaces, computational short cuts, percent, or measurement.<\/li>\n\n\n\n<li>Describe how the concept of numbers has evolved through the ages.<\/li>\n\n\n\n<li>Trace the evolution of place value systems.<\/li>\n\n\n\n<li>Describe the development of 0 from its rudimentary conceptualizations through how we write and use it today.<\/li>\n\n\n\n<li>Document the beginning of fractions and how they were used in early computations. Be sure to investigate computation involving unit fractions.<\/li>\n\n\n\n<li>Document the beginning of decimals and how they were used in early computations.<\/li>\n\n\n\n<li>Describe the development of negative numbers and how they were used in early computations.<\/li>\n\n\n\n<li>Document the beginning of complex numbers and how they were used in early computations.<\/li>\n\n\n\n<li>Document the beginning of transfinite numbers and how they were used in early computations.<\/li>\n\n\n\n<li>Investigate regular polygons. How did the names originate? How were they constructed?<\/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-5-10\" class=\"wp-block-heading\">Exercise 5.10<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Look at a recent series of mathematics texts designed for Grades 6, 7, and 8. Determine how much material is repeated from year to year. Defend why you feel this is or is not appropriate.<\/li>\n\n\n\n<li>If you could communicate with the lead editor for a middle school mathematics text publisher, what would you suggest to make the texts more appealing and useful for you, as a teacher, and for your students?<\/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--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<p class=\"wp-block-paragraph\"><strong>Computational Madness<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Simplify the following expression.<\/li>\n<\/ol>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mfrac>\n    <mn>1234567890<\/mn>\n    <mrow>\n      <msup>\n        <mn>1234567891<\/mn>\n        <mn>2<\/mn>\n      <\/msup>\n      <mo>\u2212<\/mo>\n      <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n      <mn>1234567890<\/mn>\n      <mo form=\"postfix\" stretchy=\"false\">)<\/mo>\n      <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n      <mn>1234567892<\/mn>\n      <mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo>\n    <\/mrow>\n  <\/mfrac>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">Hint:&nbsp; Try using x = 1234567891<\/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: 1234567890.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let 1234567891 = N, then 1234567890 = N \u2013 1 and 1234567892 = N + 1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mtable columnalign=\"left\">\n    <mtr>\n      <mtd class=\"tml-left\" style=\"padding:0.5ex 0pt 0.5ex 0pt;\">\n        <mrow>\n          <mfrac>\n            <mn>1234567890<\/mn>\n            <mrow>\n              <msup>\n                <mn>1234567891<\/mn>\n                <mn>2<\/mn>\n              <\/msup>\n              <mo>\u2212<\/mo>\n              <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n              <mn>1234567890<\/mn>\n              <mo form=\"postfix\" stretchy=\"false\">)<\/mo>\n              <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n              <mn>1234567892<\/mn>\n              <mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo>\n            <\/mrow>\n          <\/mfrac>\n          <mo>=<\/mo>\n        <\/mrow>\n      <\/mtd>\n    <\/mtr>\n    <mtr>\n      <mtd class=\"tml-left\" style=\"padding:0.5ex 0pt 0.5ex 0pt;\">\n        <mrow>\n          <mfrac>\n            <mrow>\n              <mi>N<\/mi>\n              <mo>\u2212<\/mo>\n              <mn>1<\/mn>\n            <\/mrow>\n            <mrow>\n              <msup>\n                <mi>N<\/mi>\n                <mn>2<\/mn>\n              <\/msup>\n              <mo>\u2212<\/mo>\n              <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n              <mi>N<\/mi>\n              <mo>\u2212<\/mo>\n              <mn>1<\/mn>\n              <mo form=\"postfix\" stretchy=\"false\">)<\/mo>\n              <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n              <mi>N<\/mi>\n              <mo>+<\/mo>\n              <mn>1<\/mn>\n              <mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo>\n            <\/mrow>\n          <\/mfrac>\n          <mo>=<\/mo>\n          <mfrac>\n            <mrow>\n              <mi>N<\/mi>\n              <mo>\u2212<\/mo>\n              <mn>1<\/mn>\n            <\/mrow>\n            <mrow>\n              <msup>\n                <mi>N<\/mi>\n                <mn>2<\/mn>\n              <\/msup>\n              <mo>\u2212<\/mo>\n              <mrow>\n                <mo fence=\"true\" form=\"prefix\">(<\/mo>\n                <msup>\n                  <mi>N<\/mi>\n                  <mn>2<\/mn>\n                <\/msup>\n                <mo>\u2212<\/mo>\n                <mn>1<\/mn>\n                <mo fence=\"true\" form=\"postfix\">)<\/mo>\n              <\/mrow>\n            <\/mrow>\n          <\/mfrac>\n          <mo>=<\/mo>\n          <mfrac>\n            <mrow>\n              <mi>N<\/mi>\n              <mo>\u2212<\/mo>\n              <mn>1<\/mn>\n            <\/mrow>\n            <mn>1<\/mn>\n          <\/mfrac>\n          <mo>=<\/mo>\n          <mn>1234567890<\/mn>\n        <\/mrow>\n      <\/mtd>\n    <\/mtr>\n  <\/mtable>\n<\/math>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<ol class=\"wp-block-list\">\n<li>By arranging the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 it is possible to come up with a fraction equivalent to one-eighth. For example:<\/li>\n<\/ol>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <mfrac>\n      <mn>1<\/mn>\n      <mn>8<\/mn>\n    <\/mfrac>\n    <mo>=<\/mo>\n    <mfrac>\n      <mn>3187<\/mn>\n      <mn>25796<\/mn>\n    <\/mfrac>\n  <\/mrow>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">Your task is to arrange the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 to come up with an equivalent fraction to one-fifth<\/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: There are many possibilities. One such answer is<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <mfrac>\n      <mn>1<\/mn>\n      <mn>5<\/mn>\n    <\/mfrac>\n    <mo>=<\/mo>\n    <mfrac>\n      <mn>52769<\/mn>\n      <mn>13845<\/mn>\n    <\/mfrac>\n  <\/mrow>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">Others include:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>&nbsp;2769 \/ 13845<\/li>\n\n\n\n<li>2973 \/ 14865<\/li>\n\n\n\n<li>9237 \/ 46185<\/li>\n\n\n\n<li>2697 \/ 13485<\/li>\n\n\n\n<li>2937 \/ 14685<\/li>\n\n\n\n<li>2967 \/ 14835<\/li>\n\n\n\n<li>3297 \/ 16485<\/li>\n\n\n\n<li>3729 \/ 18645<\/li>\n\n\n\n<li>6297 \/ 31485<\/li>\n\n\n\n<li>9723 \/ 48615<\/li>\n\n\n\n<li>9627 \/ 48135<\/li>\n\n\n\n<li>7629 \/ 38145<\/li>\n<\/ol>\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\" style=\"padding-top:0;padding-bottom:0\">\n<p class=\"wp-block-paragraph\"><strong>1.&nbsp;&nbsp; <\/strong>Create a formula that shows how to determine the number of handshakes given to any number of people. Show that your formula works by adding a column to the table in the first exercise and completing it for each of the last three entries, showing your work.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You might notice that these are the triangular numbers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The idea of generating formulas is common in the world of mathematics. We look at situations and try to come up with useful generalizations. Often these generalizations give us formulas that can be used in similar situations and that save time trying to find answers for each separate case. You have used many of these universally acceptable generalizations over your career. For example, formulas for area and perimeter, distances, rates, times, tax computations, and so on are among such generalizations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Your Turn<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Develop a formula for finding the sum of any set of consecutive counting numbers, beginning with any number.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Given that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mfrac>    <mrow>      <mrow class=\"MJX-TeXAtom-ORD\">        <mi mathvariant=\"normal\">n<\/mi>      <\/mrow>      <mo stretchy=\"false\">(<\/mo>      <mrow class=\"MJX-TeXAtom-ORD\">        <mi mathvariant=\"normal\">n<\/mi>      <\/mrow>      <mo>+<\/mo>      <mn>1<\/mn>      <mo stretchy=\"false\">)<\/mo>    <\/mrow>    <mn>2<\/mn>  <\/mfrac><\/math>&nbsp;gives the sum of the first n counting numbers, starting with 1, and that n<sup>2<\/sup> is the sum of the first n odd counting numbers, then it seems as if <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mfrac>    <mrow>      <mi>n<\/mi>      <mo stretchy=\"false\">(<\/mo>      <mi>n<\/mi>      <mo>+<\/mo>      <mn>1<\/mn>      <mo stretchy=\"false\">)<\/mo>    <\/mrow>    <mn>2<\/mn>  <\/mfrac>  <mo>\u2212<\/mo>  <msup>    <mi>n<\/mi>    <mrow class=\"MJX-TeXAtom-ORD\">      <mn>2<\/mn>    <\/mrow>  <\/msup><\/math> should give the sum of the first n even counting numbers.&nbsp;<\/li>\n<\/ul>\n\n\n\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\">    <mtr>      <mtd>        <mfrac>          <mrow>            <mi>n<\/mi>            <mo stretchy=\"false\">(<\/mo>            <mi>n<\/mi>            <mo>+<\/mo>            <mn>1<\/mn>            <mo stretchy=\"false\">)<\/mo>          <\/mrow>          <mn>2<\/mn>        <\/mfrac>        <mo>&#x2212;<!-- \u2212 --><\/mo>        <msup>          <mi>n<\/mi>          <mrow class=\"MJX-TeXAtom-ORD\">            <mn>2<\/mn>          <\/mrow>        <\/msup>      <\/mtd>      <mtd>        <mi><\/mi>        <mo>=<\/mo>        <mfrac>          <mrow>            <mi>n<\/mi>            <mo stretchy=\"false\">(<\/mo>            <mi>n<\/mi>            <mo>+<\/mo>            <mn>1<\/mn>            <mo stretchy=\"false\">)<\/mo>            <mo>&#x2212;<!-- \u2212 --><\/mo>            <mn>2<\/mn>            <msup>              <mi>n<\/mi>              <mrow class=\"MJX-TeXAtom-ORD\">                <mn>2<\/mn>              <\/mrow>            <\/msup>          <\/mrow>          <mn>2<\/mn>        <\/mfrac>      <\/mtd>    <\/mtr>    <mtr>      <mtd \/>      <mtd>        <mi><\/mi>        <mo>=<\/mo>        <mfrac>          <mrow>            <msup>              <mi>n<\/mi>              <mrow class=\"MJX-TeXAtom-ORD\">                <mn>2<\/mn>              <\/mrow>            <\/msup>            <mo>+<\/mo>            <mi>n<\/mi>            <mo>&#x2212;<!-- \u2212 --><\/mo>            <mn>2<\/mn>            <msup>              <mi>n<\/mi>              <mrow class=\"MJX-TeXAtom-ORD\">                <mn>2<\/mn>              <\/mrow>            <\/msup>          <\/mrow>          <mn>2<\/mn>        <\/mfrac>      <\/mtd>    <\/mtr>    <mtr>      <mtd \/>      <mtd>        <mi><\/mi>        <mo>=<\/mo>        <mfrac>          <mrow>            <mi>n<\/mi>            <mo>&#x2212;<!-- \u2212 --><\/mo>            <msup>              <mi>n<\/mi>              <mrow class=\"MJX-TeXAtom-ORD\">                <mn>2<\/mn>              <\/mrow>            <\/msup>          <\/mrow>          <mn>2<\/mn>        <\/mfrac>      <\/mtd>    <\/mtr>  <\/mtable><\/math>\n\n\n\n<p class=\"wp-block-paragraph\">But <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mfrac>    <mrow>      <mi>n<\/mi>      <mo>\u2212<!-- \u2212 --><\/mo>      <msup>        <mi>n<\/mi>        <mrow class=\"MJX-TeXAtom-ORD\">          <mn>2<\/mn>        <\/mrow>      <\/msup>    <\/mrow>    <mn>2<\/mn>  <\/mfrac><\/math> is negative! What is wrong?<br><br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>What famous mathematician is associated with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">  <mfrac>    <mrow>      <mrow class=\"MJX-TeXAtom-ORD\">        <mi mathvariant=\"normal\">n<\/mi>      <\/mrow>      <mo stretchy=\"false\">(<\/mo>      <mrow class=\"MJX-TeXAtom-ORD\">        <mi mathvariant=\"normal\">n<\/mi>      <\/mrow>      <mo>+<\/mo>      <mn>1<\/mn>      <mo stretchy=\"false\">)<\/mo>    <\/mrow>    <mn>2<\/mn>  <\/mfrac><\/math> which gives the sum of the first n counting numbers?<\/li>\n<\/ul>\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_6295fa1f3cd1efac74f3d3b8fcc0728e\" 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-5\/Chapter5-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_0dac62a605e8a5546bbb533176d0ddb3\" 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-5\/Chapter5-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 aria-label=\"Table of Contents\" class=\"wp-block-table-of-contents\"><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-chapter-1-exercises\">Exercises\u00a0<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-1\">Exercise 5.1<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-2\">Exercise 5.2<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-3\">Exercise 5.3<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-4\">Exercise 5.4<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-5\">Exercise 5.5<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-6\">Exercise 5.6<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-7\">Exercise 5.7<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-8\">Exercise 5.8<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-9\">Exercise 5.9<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-exercise-5-10\">Exercise 5.10<\/a><\/li><\/ol><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#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-5-skills-in-teaching-mathematics\/#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-5-skills-in-teaching-mathematics\/#aioseo-videos\">Videos<\/a><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#aioseo-problem-solving\">Problem Solving<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/#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>How wonderful it would be if we knew everything about students, teaching, mathematics, and the teaching of mathematics. We could bottle it, sell it, become rich, and solve a lot of problems for everyone in the process. We all know there is no magic formula for teaching mathematics. In the process of attempting to find [&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-171","content","type-content","status-publish","format-standard","hentry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"How wonderful it would be if we knew everything about students, teaching, mathematics, and the teaching of mathematics. We could bottle it, sell it, become rich, and solve a lot of problems for everyone in the process. We all know there is no magic formula for teaching mathematics. In the process of attempting to find\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/routledgelearning.com\/teachingsecondarymathematics\/content\/resources\/chapter-5-skills-in-teaching-mathematics\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO Pro (AIOSEO) 5.0.1.1\" \/>\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 5 \u2013 Skills in Teaching Mathematics - Teaching Secondary Mathematics\" \/>\n\t\t<meta property=\"og:description\" content=\"How wonderful it would be if we knew everything about students, teaching, mathematics, and the teaching of mathematics. We could bottle it, sell it, become rich, and solve a lot of problems for everyone in the process. 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