{"id":268,"date":"2017-08-23T04:35:31","date_gmt":"2017-08-23T04:35:31","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=268"},"modified":"2017-09-25T21:42:21","modified_gmt":"2017-09-25T21:42:21","slug":"the-effects-on-measures-of-changes-in-dimensions","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/the-effects-on-measures-of-changes-in-dimensions\/","title":{"rendered":"The Effects on Measures of Changes in Dimensions"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/measuring-other-solids\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/summarizing-data\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">The Effects on Measures of Changes in Dimensions<\/h1>\n<h4>Objective<\/h4>\n<p>In this section, we\u2019ll review measures of two- and three-dimensional figures by determining how changes in one or more of a figure\u2019s dimensions will influence those measures.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>We have reviewed various formulas used for measuring geometric figures.<\/li>\n<\/ul>\n<section>How is the area of a rectangle affected if both its length and width are doubled? How is the volume of a cube affected if the lengths of its sides are halved? In this section we\u2019ll look at a handful of measures so that you can see how problems like these are solved.How is the area of a rectangle affected if both its length and width are doubled? Well, the area of a parallelogram is given by the formula A = bh, where b is the base and h is the height.<\/p>\n<p>We need to find a way to represent the doubled length and width. (Now\u2019s a good time to note that length and width and base and height may be used interchangeably. So the area of a parallelogram is also given by the formula A = lw.)<\/p>\n<p>If l is the length, then 2l is the doubled length. Similarly, if w is the width, then 2w is the doubled width. Now just substitute the new length and new width into the formula where the old length and old width were.<\/p>\n<p>The new area is:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage025.gif\" alt=\"Solving for area of a parallelogram\" \/><\/center>The old area was lw, and the new area is 4lw. That means that if the length and width are doubled, the area is multiplied by four.<\/p>\n<p>One more? How is the volume of a cone affected if the radius is doubled and the height is divided by two? What are your initial thoughts? Do you think the volume will be affected?<\/p>\n<p>The volume of a cone is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage027.gif\" alt=\"Formula for volume of a cone\" \/>, where r is the radius of the base and h is the height. The doubled radius can be represented by 2r and the halved height can be represented by <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage030.gif\" alt=\"h\/2\" \/>. The new volume is:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage033.gif\" alt=\"Solving for volume\" \/><\/center>The old volume was <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage036.gif\" alt=\"Old solution\" \/>, and the new volume is <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage039.gif\" alt=\"New solution\" \/>. The new volume is twice the old volume.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>How is the area of an isosceles right triangle affected if the length of each leg is doubled?<\/p>\n<ol>\n<li>The area is doubled.<\/li>\n<li>The area is squared.<\/li>\n<li>The area is multiplied by 4.<\/li>\n<li>The area is multiplied by 16.<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice C is the correct answer. The area of a triangle is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage042.gif\" alt=\"Formula for area of a triangle\" \/>, where b is the base and h is the height. The base and height of an isosceles right triangle are its legs, so the area of the triangle for which the lengths of the legs are doubled is:<\/p>\n<p class=\"q-reveal center\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage045.gif\" alt=\"Solution formula\" \/><\/p>\n<p class=\"q-reveal\">The area of the old triangle was <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage047.gif\" alt=\"1\/2bh\" \/> and the area of the new triangle is 2bh. That means if the lengths of the legs of an isosceles right triangle are doubled, the area of the new triangle will be four times the area of the old triangle.<\/p>\n<\/section>\n<h3>Metric and Customary Systems of Measurement<\/h3>\n<p>The metric system is a decimal system of measurement, which means that all metric units are based on multiples of ten. For example, a kilometer is 1,000 meters and 10 millimeters equal one centimeter.<\/p>\n<p>The metric system is also called the International System of Units (abbreviated SI, for its name in French). So, metric and SI are used interchangeably.<\/p>\n<p>This table gives the most common metric prefixes and their meanings:<\/p>\n<table>\n<thead>\n<tr>\n<th>Prefix<\/th>\n<th>Symbol<\/th>\n<th>Meaning (multiply by)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilo-<\/td>\n<td>k<\/td>\n<td>1000<\/td>\n<\/tr>\n<tr>\n<td>centi-<\/td>\n<td>c<\/td>\n<td>0.01 = <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage053.gif\" alt=\"1\/100\" \/><\/td>\n<\/tr>\n<tr>\n<td>milli-<\/td>\n<td>m<\/td>\n<td>0.001 = <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage056.gif\" alt=\"1\/1000\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The metric unit of length is the meter (m); the metric unit of mass is the kilogram (kg); and the metric unit of volume is the liter (l).<\/p>\n<p>For example, 1 m = 100 cm, 1 mg = 0.001 g, and 1000 l = 1 kl.<\/p>\n<p>Although the SI unit of temperature is the Kelvin, most often, the Celsius scale is used to measure temperatures where this system is used. These are common temperatures:<\/p>\n<ul>\n<li>Water freezes at 0\u00b0 C.<\/li>\n<li>Water boils at 100\u00b0 C.<\/li>\n<\/ul>\n<h3>The Customary System<\/h3>\n<p>The <em>customary system<\/em> is the system we\u2019re accustomed to in the United States. It\u2019s also called the English system or the standard system. Customary units of length include the inch (in.), foot (ft), and mile (mi); customary units of weight include the ton (t), the pound (lb), the ounce (oz); and customary units of capacity include fluid ounces (fl. oz), pints (pt), cups (c), and quarts (qt).<\/p>\n<p>You\u2019re likely familiar with these relationships among customary units:<\/p>\n<ul>\n<li>Length: 1 mile = 5,280 feet and 1 foot = 12 inches<\/li>\n<li>Weight: 1 t = 2,000 lb and 1 lb = 16 oz<\/li>\n<li>Capacity: 1 gal = 4 qt, 1 qt = 2 pt, 1 pt = 2 c, and 1 c = 8 oz<\/li>\n<\/ul>\n<p>Most often, the Fahrenheit scale is used to measure temperatures where this system is used. These are common temperatures:<\/p>\n<ul>\n<li>Water freezes at 32\u00b0 F (which equals 0\u00b0 C).<\/li>\n<li>Water boils at 212\u00b0 F (which equals 100\u00b0 C).<\/li>\n<\/ul>\n<h3>Relating the Metric and Customary Systems<\/h3>\n<p>These facts may give you a better sense of the relationships between these systems:<\/p>\n<ul>\n<li>One liter is a little more than one quart.<\/li>\n<li>One meter is a little longer than one yard, or three feet.<\/li>\n<li>There are about 2.5 centimeters in an inch.<\/li>\n<\/ul>\n<p>And here\u2019s how we convert between the Fahrenheit and Celsius temperature scales:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage059.gif\" alt=\"Converting between Fahrenheit and Celsius\" \/><\/center>Now practice:<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Which of the following lengths is greatest?<\/p>\n<ol>\n<li>7 in<\/li>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage062.gif\" alt=\"1\/2\" \/> ft<\/li>\n<li>15 cm<\/li>\n<li>125 mm<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice A is the correct answer. Fifteen cm is about six in.; 125 mm equals 12.5 cm, which is about 5 inches; and <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage062.gif\" alt=\"1\/2\" \/> ft equals 6 in. Of the measures given, seven in. is the greatest.<\/p>\n<\/section>\n<h3>Combining Units<\/h3>\n<p>The basic units given in each of the systems above are called <em>simple units<\/em>. Simple units can be put together to derive units of measurement for other quantities. Naturally, these put-together units are called <em>derived units<\/em>.<\/p>\n<p>This may sound more complicated than it is. <em>Miles per hour<\/em> is a derived unit. So are <em>square meters<\/em>, <em>hours per week<\/em>, <em>people-hours<\/em>, and <em>kilogram-meters per second squared<\/em>. (See, some are more derived than others.) In fact, a <em>kilogram-meter per second squared<\/em> is called a Newton, so a Newton is also a derived unit, even though it looks simple.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Rate is equal to the quotient of distance and time, or <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/ratedistancetime.gif\" alt=\"Rate is distance over time\" \/>. Which of the following is a unit of rate?<\/p>\n<ol>\n<li>hours per week<\/li>\n<li>inches per foot<\/li>\n<li>miles per gallon<\/li>\n<li>meters per second<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice D is the correct answer. A unit of rate is equal to a unit of distance divided by a unit of time. A meter is a unit of distance and a second is a unit of time, so <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bimage071.gif\" alt=\"meters over seconds\" \/>, or meters per second, is a unit of rate.<\/p>\n<\/section>\n<h3>Review<\/h3>\n<ul>\n<li>The metric system is a decimal system of measurement. So metric measures of length, mass, and volume are multiples of ten of a meter, kg, and liter, respectively.<\/li>\n<li>The customary system of measurement is the system commonly used in the United States. It includes units such as inches, pounds, and gallons.<\/li>\n<li>Simple units are the most basic units of a system of measurement.<\/li>\n<li>Derived units are combinations of simple units.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/measuring-other-solids\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/summarizing-data\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 The Effects on Measures of Changes in Dimensions Objective In this section, we\u2019ll review measures of two- and three-dimensional figures by determining how changes in one or more of a figure\u2019s dimensions will influence those measures. Previously Covered We have reviewed various formulas used for measuring geometric figures. How [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-268","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/268","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=268"}],"version-history":[{"count":7,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/268\/revisions"}],"predecessor-version":[{"id":1406,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/268\/revisions\/1406"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}