{"id":9647,"date":"2012-10-01T14:29:49","date_gmt":"2012-10-01T13:29:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9647"},"modified":"2022-01-19T23:50:55","modified_gmt":"2022-01-19T23:50:55","slug":"um-quadrado-multiplicativo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9647","title":{"rendered":"Um quadrado multiplicativo"},"content":{"rendered":"<p><ul id='GTTabs_ul_9647' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9647' class='GTTabs_curr'><a  id=\"9647_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9647' ><a  id=\"9647_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_9647'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9649\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9649\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult.jpg\" data-orig-size=\"390,391\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349098483&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Quadrado multiplicativo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult.jpg\" class=\"alignright wp-image-9649\" title=\"Quadrado multiplicativo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult.jpg\" alt=\"\" width=\"300\" height=\"301\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult.jpg 390w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-150x150.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-300x300.jpg 300w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Um quadrado multiplicativo caracteriza-se por o produto dos n\u00fameros em cada linha e em cada coluna ser o mesmo.<\/p>\n<ol>\n<li>Completa o quadrado multiplicativo.<\/li>\n<li>No quadrado obtido, multiplica cada n\u00famero por $-3$. O que observas?<\/li>\n<li>Se adicionares $-2$ aos n\u00fameros do quadrado obtido em 1., o que observas?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9647' onClick='GTTabs_show(1,9647)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9647'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9652\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9652\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol.jpg\" data-orig-size=\"390,390\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349098530&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Quadrado multiplicativo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol.jpg\" class=\"alignright wp-image-9652\" title=\"Quadrado multiplicativo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol.jpg 390w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol-150x150.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/quadadomult-sol-300x300.jpg 300w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Podemos determinar o valor do produto dos n\u00fameros em cada linha e em cada coluna utilizando a 1.\u00aa coluna do quadrado: $$5 \\times \\left( { &#8211; 4} \\right) \\times 4 =\u00a0 &#8211; 20 \\times 4 =\u00a0 &#8211; 80$$<br \/>\nDe seguida, calculam-se os valores em falta em cada um dos quadradinhos:<br \/>\n$$\\begin{array}{*{20}{l}} {5 \\times 8 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{40 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{\\left( ? \\right) =\u00a0 &#8211; 2}\\\\ { &#8211; 4 \\times \\left( ? \\right) \\times 2 =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{ &#8211; 8 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{\\left( ? \\right) = 10}\\\\ {8 \\times 10 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{80 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{\\left( ? \\right) =\u00a0 &#8211; 1}\\\\ { &#8211; 2 \\times 2 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{ &#8211; 4 \\times \\left( ? \\right) =\u00a0 &#8211; 80}&amp; \\Rightarrow &amp;{\\left( ? \\right) = 20} \\end{array}$$<\/li>\n<li>Multiplicando por $-3$ cada um dos n\u00fameros do quadrado obtido em 1., obt\u00e9m-se o seguinte quadrado:<br \/>\n<table class=\" aligncenter\" style=\"width: 80%; height: 120px;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-15$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-24$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$6$<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$12$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-30$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-6$<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-12$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$3$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-60$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Podemos\u00a0concluir\u00a0que, agora, o produto dos n\u00fameros em cada linha e em cada coluna \u00e9 $ &#8211; 80 \\times \\left[ {\\left( { &#8211; 3} \\right) \\times \\left( { &#8211; 3} \\right) \\times \\left( { &#8211; 3} \\right)} \\right] =\u00a0 &#8211; 80 \\times \\left( { &#8211; 27} \\right) = 2160$, pois cada um dos valores foi multiplicado por $-3$.<\/p>\n<p>Estamos ainda face a um quadrado multiplicativo. Com efeito, temos:<br \/>\n$$\\begin{array}{*{20}{l}} {Linha}&amp;{}&amp;{Coluna}\\\\ { &#8211; 15 \\times \\left( { &#8211; 24} \\right) \\times 6 = 360 \\times 6 = 2160}&amp;{}&amp;{ &#8211; 15 \\times 12 \\times \\left( { &#8211; 12} \\right) =\u00a0 &#8211; 180 \\times \\left( { &#8211; 12} \\right) = 2160}\\\\ {12 \\times \\left( { &#8211; 30} \\right) \\times \\left( { &#8211; 6} \\right) = \\left( { &#8211; 360} \\right) \\times \\left( { &#8211; 6} \\right) = 2160}&amp;{}&amp;{ &#8211; 24 \\times \\left( { &#8211; 30} \\right) \\times 3 = 720 \\times 3 = 2160}\\\\ { &#8211; 12 \\times 3 \\times \\left( { &#8211; 60} \\right) =\u00a0 &#8211; 36 \\times \\left( { &#8211; 60} \\right) = 2160}&amp;{}&amp;{6 \\times \\left( { &#8211; 6} \\right) \\times \\left( { &#8211; 60} \\right) =\u00a0 &#8211; 36 \\times \\left( { &#8211; 60} \\right) = 2160} \\end{array}$$\u00ad<\/p>\n<\/li>\n<li>\n<p>Adicionando $-2$ cada um dos n\u00fameros do quadrado obtido em 1., obt\u00e9m-se o seguinte quadrado:<\/p>\n<table class=\" aligncenter\" style=\"width: 80%; height: 100px;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$3$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$6$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-4$<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-6$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$8$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$0$<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$2$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$-3$<\/td>\n<td style=\"border: 1px solid #cc0000; text-align: center;\">$18$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Podemos concluir que j\u00e1 n\u00e3o estamos face a um quadrado multiplicativo. (PORQU\u00ca?)<\/p>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9647' onClick='GTTabs_show(0,9647)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um quadrado multiplicativo caracteriza-se por o produto dos n\u00fameros em cada linha e em cada coluna ser o mesmo. Completa o quadrado multiplicativo. No quadrado obtido, multiplica cada n\u00famero por $-3$.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20704,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[317,97,318],"tags":[428,320,319],"series":[],"class_list":["post-9647","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-7-o-ano","category-aplicando","category-numeros-inteiros","tag-7-o-ano","tag-multiplicacao","tag-numeros-inteiros-2"],"views":2507,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/7V1Pag014-10_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9647","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9647"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9647\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20704"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9647"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9647"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9647"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}