{"id":3940,"date":"2022-05-12T12:25:11","date_gmt":"2022-05-12T19:25:11","guid":{"rendered":"https:\/\/mathtalk.sdsu.edu\/wordpress\/?page_id=3940"},"modified":"2023-11-09T17:32:03","modified_gmt":"2023-11-10T01:32:03","slug":"algebraic-expressions-lesson-6-teachers","status":"publish","type":"page","link":"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/algebraic-expressions-lesson-6-teachers\/","title":{"rendered":"Algebraic Expressions Lesson 6 (Teachers)"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\"><strong><strong><strong><strong><strong>Comparing Quadratic Relationships Using Visual Patterns<\/strong><\/strong><\/strong><\/strong><\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Haleemah and ET explore the context of cobblestone patterns from the Czech Republic. They make sense of another student, Amir\u2019s, way of seeing the relationship between a Size&nbsp;Number and the Haleemah and ET are given another way of seeing a pattern in a cobblestone figure, this time starting with an algebraic equation. Given Nicole\u2019s algebraic equation, the students will work to make sense of how she may have been seeing the relationship between the size number and the number of gray stones. Then, they will compare and explore the equality between Amir and Nicole\u2019s methods.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\"><a href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/algebraic-expressions-lesson-6-teachers\/algebraic-expressions-lesson-6-episode-1-teachers\/\">Episode 1: Making Sense<\/a><\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">ET and Haleemah make sense of the cobblestone context and think ET and Haleemah make sense of Nicole\u2019s algebraic and arithmetic equations and create a drawing that they believe describes how she was seeing the pattern.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\"><a href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/algebraic-expressions-lesson-6-teachers\/algebraic-expressions-lesson-6-episode-2-teachers\/\">Episode 2: Exploring<\/a><\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">The students apply Nicole\u2019s method for finding the total number of gray cobblestones for a new&nbsp;Size&nbsp;Number.&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<h5 class=\"wp-block-heading\"><a href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/algebraic-expressions-lesson-6-teachers\/algebraic-expressions-lesson-6-episode-3-teachers\/\">Episode 3: Reflecting<\/a><\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">ET and Haleemah reflect on their generalized algebraic expression and make sense of each portion of their expression in the cobblestone context in two ways.&nbsp;<\/p>\n\n\n\n<h5 class=\"wp-block-heading\"><a href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/algebraic-expressions-lesson-6-teachers\/algebraic-expressions-lesson-6-episode-4-teachers\/\">Episode 4: Exploring<\/a><\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">Haleemah and ET explore the equality between Amir and Nicole\u2019s methods by discussing the similarities and resolving the differences between the two methods.&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<h5 class=\"wp-block-heading\"><a href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/algebraic-expressions-lesson-6-teachers\/algebraic-expressions-lesson-6-episode-5-teachers\/\">Episode 5: Reflecting<\/a><\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">The students look at someone else\u2019s reasoning about a key difference between Amir\u2019s and Nicole\u2019s methods. They use this to explain what parts of the algebra equations mean in the cobblestone pattern.&nbsp;&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Mathematics in this Lesson<\/h3>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Targeted Understandings <input type='hidden' bg_collapse_expand='6a249e483f6534087867656' value='6a249e483f6534087867656'><input type='hidden' id='bg-show-more-text-6a249e483f6534087867656' value=' '><input type='hidden' id='bg-show-less-text-6a249e483f6534087867656' value=' '><button id='bg-showmore-action-6a249e483f6534087867656' class='bg-showmore-plg-button bg-blue-button bg-arrow '   style=\" color:white;\"> <\/button><div id='bg-showmore-hidden-6a249e483f6534087867656' ><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This lesson can help students: <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Make connections between drawings, verbal generalizations, and algebraic expressions and be able to reverse these connections (e.g., interpret an algebraic equation in terms of a visual pattern in a drawing).\u00a0<\/li>\n\n\n\n<li>Develop\u00a0quantitative meaning for the distributive property of multiplication over addition as used in algebraic expressions; for example, understand why x(x + 1) = x<sup>2<\/sup>\u00a0+ x and why x(x + 1) \u2260 x<sup>2<\/sup>\u00a0+ 1.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/div><\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Common Core Math Standards <input type='hidden' bg_collapse_expand='6a249e483f6f78006335017' value='6a249e483f6f78006335017'><input type='hidden' id='bg-show-more-text-6a249e483f6f78006335017' value=' '><input type='hidden' id='bg-show-less-text-6a249e483f6f78006335017' value=' '><button id='bg-showmore-action-6a249e483f6f78006335017' class='bg-showmore-plg-button bg-blue-button bg-arrow '   style=\" color:white;\"> <\/button><div id='bg-showmore-hidden-6a249e483f6f78006335017' ><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.thecorestandards.org\/Math\/Content\/HSA\/SSE\/B\/3\/\"><strong>CCSS.M.HSA.SSE.B.3<\/strong><\/a><strong>.&nbsp;<\/strong><em>Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.<\/em><br><br>In the previous lesson, ET and Haleemah examined, utilized, and explained features of an equation for the total number of gray stones in a growing cobblestone pattern. In this lesson, they work with a different equation, one created by a new fictional student named Nicole. First, they use specific examples of the cobblestone pattern (Size 3 and Size 9) to make sense of Nicole\u2019s equation, connecting expressions in the equation to features of the visual pattern&nbsp;<strong>[Episode 1, 1:52; Episode 3]<\/strong>. They move beyond this empirical approach when comparing Nicole\u2019s equation to Amir\u2019s equation. In addition to noting that the two equations produce the same total given the same input&nbsp;<strong>[Episode 4, 1:30]<\/strong>, they also analyze algebraic expressions across the two equations&nbsp;<strong>[Episode 4, 4:39]<\/strong>&nbsp;and reason about how those expressions are equivalent by linking them to the same quantities in the cobblestone context&nbsp;<strong>[Episode 4, 7:10]<\/strong>.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.thecorestandards.org\/Math\/Content\/HSA\/SSE\/A\/2\/\"><strong>CCSS.M.HSA.SSE.A.2<\/strong><\/a><strong>.\u00a0<\/strong><em>Use the structure of an expression to identify ways to rewrite it.<\/em><br><br>As part of this lesson, Haleemah and ET must make sense of a new equation for the cobblestone pattern context. Haleemah and ET claim that one of the expressions, x \u2022 (x + 1), is equivalent to the expression x \u2022 x + x. They initially justify this by examining a specific instance for x = 5\u00a0<strong>[Episode 4, 7:10]<\/strong>. Later, they reason more abstractly by examining the structure of the cobblestone pattern and linking it to the expressions. They also consider the thinking of another student, who claims (incorrectly) that\u00a0x\u00a0\u2022\u00a0(x + 1) = x<sup>2<\/sup>\u00a0+ 1\u00a0<strong>[Episode 5]<\/strong>. Haleemah and ET note that, in the cobblestone context,\u00a0x<sup>2<\/sup>\u00a0+ 1\u00a0does\u00a0<em>not<\/em>\u00a0represent the number of stones in the middle section because it omits some of the stones in the extra column<strong>\u00a0[Episode 5, 5:49]<\/strong>. In doing so, they provide another justification for why\u00a0x\u00a0\u2022\u00a0(x + 1)\u00a0can be rewritten as x \u2022 x + x or\u00a0x<sup>2<\/sup>\u00a0+ x.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/div><\/p>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\">Common Core Math Practices <input type='hidden' bg_collapse_expand='6a249e483f7cc7062182115' value='6a249e483f7cc7062182115'><input type='hidden' id='bg-show-more-text-6a249e483f7cc7062182115' value=' '><input type='hidden' id='bg-show-less-text-6a249e483f7cc7062182115' value=' '><button id='bg-showmore-action-6a249e483f7cc7062182115' class='bg-showmore-plg-button bg-blue-button bg-arrow '   style=\" color:white;\"> <\/button><div id='bg-showmore-hidden-6a249e483f7cc7062182115' ><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/www.thecorestandards.org\/Math\/Practice\/MP3\/\"><strong>CCSS.MATH PRACTICE.MP3<\/strong><\/a>.&nbsp;<em>Construct viable arguments and critique the reasoning of others.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">According to the CCSSM, \u201cMathematically proficient students\u2026can recognize and use counterexamples\u2026 justify their conclusions, communicate them to others, and respond to the arguments of others. They\u2026distinguish correct logic or reasoning from that which is flawed, and\u2014if there is a flaw in an argument\u2014explain what it is.\u201d In this lesson, Haleemah and ET construct many viable arguments. In the first part of this lesson, they support their claim that Nicole\u2019s equation is valid for the cobblestone pattern, first by examining the equation for specific cases\u00a0<strong>[e.g., Episode 1:44; Episode 2, 2:43]<\/strong>, and later by explaining how each expression in the equation matches features of the cobblestone pattern\u00a0<strong>[e.g., Episode 3, 0:52]<\/strong>. They also critique the reasoning of others. Their arguments about Nicole\u2019s equation serve as analysis of Nicole\u2019s assumed reasoning, with Haleemah and ET evaluating that reasoning as valid and accurate. Moreover, they critique the reasoning of another friend who demonstrates incorrect reasoning. For this friend, ET and Haleemah first offer a counterexample for why\u00a0x \u2022 (x + 1) \u2260 x<sup>2<\/sup>\u00a0+ 1, showing that the two expressions are not equal for when x = 5\u00a0<strong>[Episode 5, 1:52]<\/strong>. They then reason more abstractly by linking the algebraic expressions directly to the visual features of the cobblestone pattern to show why the two expressions are unequal.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/div><\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/\" style=\"background-color:#2d4059\">Home<\/a><\/div>\n\n\n\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/\" style=\"background-color:#2d4059\">Teachers Home<\/a><\/div>\n\n\n\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/mathtalk.sdsu.edu\/wordpress\/mathtalk-for-teachers\/meaning-and-equivalence-of-algebraic-expressions-unit-teachers\/\" style=\"background-color:#2d4059\">ALGEBRAIC EXPRESSIONS<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Comparing Quadratic Relationships Using Visual Patterns Haleemah and ET explore the context of cobblestone patterns from the Czech Republic. They make sense of another student, Amir\u2019s, way of seeing the relationship between a Size&nbsp;Number and the Haleemah and ET are given another way of seeing a pattern in a cobblestone figure, this time starting with [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":549,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-3940","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/pages\/3940","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/comments?post=3940"}],"version-history":[{"count":9,"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/pages\/3940\/revisions"}],"predecessor-version":[{"id":5320,"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/pages\/3940\/revisions\/5320"}],"up":[{"embeddable":true,"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/pages\/549"}],"wp:attachment":[{"href":"https:\/\/mathtalk.sdsu.edu\/wordpress\/wp-json\/wp\/v2\/media?parent=3940"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}