{"id":890,"date":"2017-05-08T03:47:58","date_gmt":"2017-05-08T03:47:58","guid":{"rendered":"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/?page_id=890"},"modified":"2017-10-27T15:20:44","modified_gmt":"2017-10-27T19:20:44","slug":"early-algebra","status":"publish","type":"page","link":"https:\/\/investigations.terc.edu\/inv2\/curriculum-in-the-classroom\/early-algebra\/","title":{"rendered":"Early Algebra"},"content":{"rendered":"<p>[et_pb_section fb_built=&#8221;1&#8243; admin_label=&#8221;section&#8221; _builder_version=&#8221;3.0.47&#8243;][et_pb_row admin_label=&#8221;row&#8221; _builder_version=&#8221;3.0.47&#8243; background_size=&#8221;initial&#8221; background_position=&#8221;top_left&#8221; background_repeat=&#8221;repeat&#8221; _i=&#8221;0&#8243; _address=&#8221;0.0&#8243;][et_pb_column type=&#8221;4_4&#8243; _builder_version=&#8221;3.0.47&#8243; parallax=&#8221;off&#8221; parallax_method=&#8221;on&#8221;][et_pb_text admin_label=&#8221;Page Title&#8221; _builder_version=&#8221;3.0.76&#8243; background_size=&#8221;initial&#8221; background_position=&#8221;top_left&#8221; background_repeat=&#8221;repeat&#8221; module_alignment=&#8221;left&#8221;]<\/p>\n<div id=\"header_bar\">\n<h1>Early Algebra: Generalized Arithmetic in the Classroom<\/h1>\n<\/div>\n<p>[\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row _builder_version=&#8221;3.0.47&#8243; background_size=&#8221;initial&#8221; background_position=&#8221;top_left&#8221; background_repeat=&#8221;repeat&#8221; _i=&#8221;1&#8243; _address=&#8221;0.1&#8243;][et_pb_column type=&#8221;2_3&#8243; _builder_version=&#8221;3.0.47&#8243; parallax=&#8221;off&#8221; parallax_method=&#8221;on&#8221;][et_pb_text _builder_version=&#8221;3.0.76&#8243; background_size=&#8221;initial&#8221; background_position=&#8221;top_left&#8221; background_repeat=&#8221;repeat&#8221; module_alignment=&#8221;left&#8221;]<\/p>\n<p>Embedded in the number and operations units at each grade is the early algebra work focused on generalizations that arise in the course of students\u2019 study of counting, numbers and operations. In mathematics a generalization is a general claim you can make about the way numbers or operations work. An example of a generalization is: in addition you can change the order of the addends and the sum remains the same.<\/p>\n<p><strong>Implicit Generalizations<\/strong><\/p>\n<p>What does it look like for students to work on generalizations? Generalizations are implicit in the strategies students use to solve computation problems. Students may not be able to articulate these generalizations, but they use their understanding of number relationships and the operations as they solve problems.<\/p>\n<p>For example, in the following video, students share strategies for solving a story problem about 6 cars plus 7 cars.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/player.vimeo.com\/video\/54622806?title=0&amp;byline=0&amp;portrait=0\" width=\"455\" height=\"335\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<table border=\"0\" width=\"474\">\n<tbody>\n<tr>\n<td width=\"340\"><\/td>\n<td align=\"right\" width=\"121\"><a href=\"https:\/\/investigations.terc.edu\/inv2\/curriculum-by-grade-level\/grade1-curriculum\/\">All Gr. 1 videos<\/a>.<a href=\"https:\/\/investigations.terc.edu\/inv2\/curriculum-by-grade-level\/grade1-curriculum\/\"><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In order for Elizabeth to use 6 + 6 to help her solve 6 + 7, she needed to know that 6 + 6 plus 1 more equals 6 + 7 and that if she added 6 + (6+1), the sum would go up by 1. A generalization implicit in her solution is: if you add an amount to an addend in an addition problem, the sum will increase by that same amount.<\/p>\n<p><img src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/04\/green_questionmark.jpg\" align=\"left\" \/>Derek used a different generalization. What did Derek do? What did he understand? How would you put the generalization implicit in his solution into words? You might start with: \u201cIn addition, if you\u2026\u201d<\/p>\n<p><strong>Articulating and Justifying Generalizations<\/strong><\/p>\n<p>At each grade level there are explicit opportunities for students to articulate, investigate and justify some of the generalizations that are often implicit in their computational strategies.\u00a0Teachers help students articulate and justify these generalizations by asking questions such as:\u00a0Would that work with other numbers?\u00a0 Would it always work? How do you know? They ask students to use representations, models and contexts to investigate and justify their claims.<\/p>\n<p>In grade 2 students make generalizations about what happens when you add two even numbers, two odd numbers, or an even and an odd number. Students are asked to justify their ideas.<\/p>\n<p>Consider the representations students use to show why an even plus an even number is an even number.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-medium wp-image-894\" src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page45-300x175.jpg\" alt=\"\" width=\"300\" height=\"175\" srcset=\"https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page45-300x175.jpg 300w, https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page45.jpg 768w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-medium wp-image-892\" src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page.44-300x242.jpg\" alt=\"\" width=\"300\" height=\"242\" srcset=\"https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page.44-300x242.jpg 300w, https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page.44.jpg 384w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>They use similar representations to show why an odd number plus an odd number is an even number:<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-medium wp-image-893\" src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page42greensquares-300x204.jpg\" alt=\"\" width=\"300\" height=\"204\" srcset=\"https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page42greensquares-300x204.jpg 300w, https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2_U8page42greensquares.jpg 384w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-medium wp-image-895\" src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/05\/2U8p42-224x300.jpg\" alt=\"\" width=\"224\" height=\"300\" srcset=\"https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2U8p42-224x300.jpg 224w, https:\/\/investigations.terc.edu\/inv2\/wp-content\/uploads\/2017\/05\/2U8p42.jpg 234w\" sizes=\"(max-width: 224px) 100vw, 224px\" \/><br \/> <img src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/04\/green_questionmark.jpg\" align=\"left\" \/>How do these representations justify the generalizations students make about adding even and odd numbers?\u00a0What do the representations show about what these students understand about even and odd numbers? about addition?<\/p>\n<p><strong>Applying Generalizations<\/strong><\/p>\n<p>Once students have articulated and justified a generalization, they can use it to help them solve computation problems. In the following video 3rd grade students use a generalization they have already worked on, which they call the \u201cplus\/minus rule\u201d, to solve 623 + 249.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/player.vimeo.com\/video\/35530463?title=0&amp;byline=0&amp;portrait=0\" width=\"455\" height=\"335\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<table border=\"0\" width=\"471\">\n<tbody>\n<tr>\n<td width=\"340\"><\/td>\n<td align=\"right\" width=\"121\"><a href=\"https:\/\/investigations.terc.edu\/inv2\/curriculum-by-grade-level\/grade3-curriculum\/\">All Gr. 3 videos<\/a>.<a href=\"https:\/\/investigations.terc.edu\/inv2\/curriculum-by-grade-level\/grade3-curriculum\/\"><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img src=\"http:\/\/70.40.220.109\/~tercedu\/investigations\/inv2\/wp-content\/uploads\/2017\/04\/green_questionmark.jpg\" align=\"left\" \/>How might you put the \u201cplus\/minus rule\u201d into words?\u00a0 You might start with: \u201cIn addition, if you\u2026\u201d How does Kiara show what she understands about this generalization?\u00a0 How does Rose apply this generalization to the problem: 623 + 249?\u00a0 How does Kiara apply it?<\/p>\n<p><span style=\"border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px\/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: #bd081c no-repeat scroll 3px 50% \/ 14px 14px; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer;\">Save<\/span><\/p>\n<p>[\/et_pb_text][\/et_pb_column][et_pb_column type=&#8221;1_3&#8243; _builder_version=&#8221;3.0.47&#8243; parallax=&#8221;off&#8221; parallax_method=&#8221;on&#8221;][et_pb_cta title=&#8221;Learn More&#8221; use_background_color=&#8221;off&#8221; background_layout=&#8221;light&#8221; admin_label=&#8221;Learn More&#8221; _builder_version=&#8221;3.0.76&#8243; header_text_color=&#8221;#005285&#8243; background_size=&#8221;initial&#8221; background_position=&#8221;top_left&#8221; background_repeat=&#8221;repeat&#8221; use_border_color=&#8221;on&#8221; border_color=&#8221;#005285&#8243; text_orientation=&#8221;left&#8221; custom_padding=&#8221;20px|20px|20px|20px&#8221;]<\/p>\n<p><a href=\"http:\/\/investigations.terc.edu\/inv2\/curriculum-features\/early-algebra\/\">Explore the features of the curriculum that highlight the early algebra<\/a> in Investigations, and support teachers in pursuing them.<\/p>\n<p><strong>Additional Resources<\/strong><\/p>\n<ul>\n<li><a href=\"http:\/\/www.heinemann.com\/products\/E04191.aspx\" target=\"\u201d_blank\u201d\"><em>Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades<\/em><\/a> by Susan Jo Russell et. al.<\/li>\n<li><a href=\"http:\/\/www.heinemann.com\/products\/E00565.aspx\" target=\"\u201d_blank\u201d\"><em>Thinking Mathematically: Integrating Arithmetic &#038; Algebra in Elementary School<\/em><\/a> by Thomas Carpenter et. al.<\/li>\n<li><a href=\"http:\/\/www.routledge.com\/books\/details\/9780805854732\/\" target=\"\u201d_blank\u201d\"><em>Algebra in the Early Grades<\/em><\/a> by James J. Kaput et. al.<\/li>\n<\/ul>\n<p>[\/et_pb_cta][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Early Algebra: Generalized Arithmetic in the Classroom Embedded in the number and operations units at each grade is the early algebra work focused on generalizations that arise in the course of students\u2019 study of counting, numbers and operations. In mathematics a generalization is a general claim you can make about the way numbers or operations [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":490,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"","_et_gb_content_width":"","_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/PawyiF-em","_links":{"self":[{"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/pages\/890"}],"collection":[{"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/comments?post=890"}],"version-history":[{"count":6,"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/pages\/890\/revisions"}],"predecessor-version":[{"id":1712,"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/pages\/890\/revisions\/1712"}],"up":[{"embeddable":true,"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/pages\/490"}],"wp:attachment":[{"href":"https:\/\/investigations.terc.edu\/inv2\/wp-json\/wp\/v2\/media?parent=890"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}