{"id":262,"date":"2017-08-23T04:34:19","date_gmt":"2017-08-23T04:34:19","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=262"},"modified":"2017-09-25T21:08:36","modified_gmt":"2017-09-25T21:08:36","slug":"representational-systems","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/representational-systems\/","title":{"rendered":"Representational Systems"},"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\/proofs\">\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\/measuring-triangles\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Representational Systems<\/h1>\n<h4>Objective:<\/h4>\n<p>In this lesson you will learn to perform a variety of computations involving basic geometric figures and their parts.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The slope of a line describes the steepness of the line. Two lines are parallel if they have the same slope. Two lines are perpendicular if the slope of each is the negative reciprocal of the slope of the other.<\/li>\n<li>A parallelogram is a quadrilateral whose opposite sides are parallel. A rectangle is a parallelogram with four right angles.<\/li>\n<\/ul>\n<section>\n<h3>The Coordinate Plane<\/h3>\n<p>A coordinate plane is formed by two perpendicular number lines that intersect at their points of origin.<\/p>\n<p>The location of any point in the coordinate plane is given by an ordered pair of the form (x,y). It\u2019s called ordered, because the order matters: the first value in the pair is the x-coordinate of the point and the second value is the y-coordinate.<\/p>\n<p>The figure below shows a <strong>square<\/strong> in the coordinate plane.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/square-in-cp.jpg\" alt=\"Coordinate plane\" \/><\/center><\/p>\n<p class=\"figcaption\">Coordinate Plane<\/p>\n<p>But how can we be sure it\u2019s a square? Well, we must consider what we know about squares.<\/p>\n<ul>\n<li>A square is a quadrilateral, so it must have four sides.<\/li>\n<li>A square is a parallelogram, so its opposite sides must be parallel.<\/li>\n<li>A square is a rectangle, so it must have four right angles.<\/li>\n<li>A square is a rhombus, so all of its sides must be congruent.<\/li>\n<\/ul>\n<p>Let\u2019s check each of these to confirm that ABCD is a square. Clearly, it has four sides but are the opposite sides parallel? Sides AB and CD have the same steepness, so they\u2019re parallel. In fact, they\u2019re also parallel to the x-axis. Sides BC and AD also have the same steepness, so they\u2019re parallel, as well. And they\u2019re parallel to the y-axis. Thus, quadrilateral ABCD is a parallelogram.<\/p>\n<p>The opposite angles of a parallelogram are always congruent, so now we only need to check if angles DAB and ABC are right angles. We know that perpendicular lines form right angles, so if side AD is perpendicular to side AB, and if side AB is perpendicular to side BC, then all of the angles must be right angles. We can check this using a <abbr title=\"A semicircular instrument for measuring and constructing angles.\">protractor<\/abbr> to measure the angles.<\/p>\n<p>Here\u2019s a close-up of angle DAB and the protractor with which we\u2019ll measure it.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3protractor.jpg\" alt=\"Protractor\" \/><\/center><\/p>\n<p class=\"figcaption\">Protractor<\/p>\n<p>To measure the angle:<\/p>\n<p>Find the center hole on the straight edge of the protractor. Place the hole over the vertex of the angle.<\/p>\n<ul>\n<li>Line up the zero on the straight edge of the protractor with one of the sides of the angle.<\/li>\n<li>Find the point where the second side of the angle intersects the curved edge of the protractor.<\/li>\n<li>Read the number that is written on the inside scale of the protractor at the point of intersection. This is the measure of the angle in degrees.<\/li>\n<\/ul>\n<p>If the side of the angle were aligned with the left side of the straight edge of the protractor, you would have read the measure of the angle from the outside scale.<\/p>\n<p>This angle measures 90\u00b0, so it is indeed a right angle. We could repeat this process to measure angle ABC to find out that it is a right angle, too.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the measure in degrees of angle JKL?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3protractor2.jpg\" alt=\"Angle JKL on protractor\" \/><\/center><\/p>\n<ol>\n<li>55\u00b0<\/li>\n<li>60\u00b0<\/li>\n<li>125\u00b0<\/li>\n<li>130\u00b0.<\/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. Since the side of the angle is aligned with the right side of the straight edge, use the inside scale to read the angle\u2019s measure in degrees. The measures increase counterclockwise, so the angle\u2019s measure is between 120\u00b0 and 130\u00b0. The measure of the angle is 125\u00b0.<\/p>\n<\/section>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/square-in-cp.jpg\" alt=\"Coordinate plane\" \/><\/center><\/p>\n<p class=\"figcaption\">Coordinate Plane<\/p>\n<p>Now we can continue to verify that parallelogram ABCD above is a square. So far we know:<\/p>\n<ul>\n<li>It has four sides.<\/li>\n<li>Its opposite sides are parallel.<\/li>\n<li>It has four right angles.<\/li>\n<\/ul>\n<p>All we need to do now is measure the lengths of the sides. The opposite sides of a parallelogram are congruent, so all we need to do is measure two adjacent sides.<\/p>\n<p>We\u2019ll measure the lengths of <span class=\"line\">AB<\/span> and <span class=\"line\">BC<\/span>. The length of a horizontal line segment equals the difference between the x-coordinates. The endpoints of <span class=\"line\">AB<\/span> are A (\u20131, 3) and B (3, 3), so the difference between the x-coordinates is 3 \u2013 (\u20131), or 4. Distance must be positive, so if we had found the difference to be \u20131 \u2013 (3), or \u20134, we would still call the distance 4. So <span class=\"line\">AB<\/span> = 4. Thus, <span class=\"line\">CD<\/span> is also equal to 4.<\/p>\n<p>The length of a vertical line segment equals the difference between the y-coordinates. The endpoints of <span class=\"line\">BC<\/span> are B (3, 3) and C (3, \u20131), so the difference between the y-coordinates is \u20131 \u2013 (3), or \u20134.<\/p>\n<p>So <span class=\"line\">BC<\/span> = 4. Thus <span class=\"line\">AD<\/span>, is also equal to 4.<\/p>\n<p>We\u2019ve shown that all sides of ABCD have the same length, so they\u2019re all congruent. Therefore, ABCD is a square.<\/p>\n<h3>Review<\/h3>\n<ul>\n<li>A coordinate plane is formed by two perpendicular number lines that intersect at their points of origin.<\/li>\n<li>You can use a coordinate plane to measure geometric figures.<\/li>\n<li><abbr title=\"A semicircular instrument for measuring and constructing angles.\">Protractors<\/abbr> are used to measure angles.<\/li>\n<li>The length of a horizontal line segment equals the difference between the x-coordinates. The length of a vertical line segment equals the difference between the y-coordinates.<\/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\/proofs\">\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\/measuring-triangles\">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 Representational Systems Objective: In this lesson you will learn to perform a variety of computations involving basic geometric figures and their parts. Previously Covered: The slope of a line describes the steepness of the line. Two lines are parallel if they have the same slope. Two lines are perpendicular [&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-262","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/262","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=262"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/262\/revisions"}],"predecessor-version":[{"id":1393,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/262\/revisions\/1393"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}