{"id":2173,"date":"2022-05-31T07:00:07","date_gmt":"2022-05-31T12:00:07","guid":{"rendered":"https:\/\/mathisvisual.com\/?p=2173"},"modified":"2022-11-22T11:17:16","modified_gmt":"2022-11-22T16:17:16","slug":"linear-equations-with-graphs","status":"publish","type":"post","link":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/","title":{"rendered":"Constructing Linear Equations From A Graph With Rise Over Run"},"content":{"rendered":"\r\n<p><iframe loading=\"lazy\" title=\"Constructing Linear Equations From A Graph With Rise Over Run | Visual Number Talk Prompts\" width=\"1104\" height=\"621\" src=\"https:\/\/www.youtube.com\/embed\/RpV5h6nDjLE?wmode=transparent&amp;rel=0&amp;feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\r\n<h2><b>In This Set of Visual Number Talk Prompts\u2026<\/b><\/h2>\r\n<p><span style=\"font-weight: 400;\">Students will extend shrinking linear patterns when given the graph of a line. They will make near and far predictions and describe the pattern in words.\u00a0<\/span><\/p>\r\n<h2><b>Intentionality\u2026<\/b><\/h2>\r\n<p><span style=\"font-weight: 400;\">Students will engage in a visual math talk using graphical representations of shrinking patterns and will have an opportunity to complete independent purposeful practice. The math talk serves to develop a deeper understanding of the following big ideas:\u00a0<\/span><\/p>\r\n<ul>\r\n<li><span style=\"font-weight: 400;\">Patterns can be extended because they are repetitive by nature.<\/span><\/li>\r\n<li><span style=\"font-weight: 400;\">Pattern rules are generalizations about a pattern, and they can be described in words.<\/span><\/li>\r\n<li><span style=\"font-weight: 400;\">A pattern can be extended to make a prediction.\u00a0<\/span><\/li>\r\n<li><span style=\"font-weight: 400;\">For far predictions, calculations are required for efficiency.\u00a0<\/span><\/li>\r\n<li><span style=\"font-weight: 400;\">In a shrinking pattern, the values increase.<\/span><\/li>\r\n<li><span style=\"font-weight: 400;\">In a linear pattern, the values increase\/decrease at the same rate.<\/span><\/li>\r\n<li><span style=\"font-weight: 400;\">Graphical representations of linear growing and shrinking patterns appear as straight lines.<\/span><\/li>\r\n<\/ul>\r\n<h2>String of Related Problems<\/h2>\r\n<p><span style=\"font-weight: 400;\">In today\u2019s set of <\/span><b>visual math talk prompts<\/b><span style=\"font-weight: 400;\">, students will be presented with various graphical representations of linear patterns, one at a time. Ask students to tell you what they know about the pattern and to jot down everything that they know based on what they see.\u00a0<\/span><\/p>\r\n<p><span style=\"font-weight: 400;\">After students have an opportunity to turn and talk as well as share out to the group as a whole, students will be asked to answer the following:<\/span><\/p>\r\n<ol>\r\n<li><b><i>What type of pattern is this, how do you know?<\/i><\/b><\/li>\r\n<li><b><i>What context could be represented by this pattern?<\/i><\/b><\/li>\r\n<li><b><i>What is the pattern rule?<\/i><\/b><\/li>\r\n<li><b><i>Predict another value in this pattern not represented by this graph. Defend your answer.\u00a0<\/i><\/b><\/li>\r\n<\/ol>\r\n<p><span style=\"font-weight: 400;\">Since today\u2019s set of visual patterns (geometric patterns) are linear relationships, we will be intentionally seeking opportunities to identify and highlight the <\/span><b><i>initial value <\/i><\/b><span style=\"font-weight: 400;\">(or y-intercept) and the <\/span><b><i>rate of change <\/i><\/b><span style=\"font-weight: 400;\">(or slope of the line) from the given graphical representations by leveraging the equivalent ratios that reveal the rate when divided partitively. This idea is often referred to as <\/span><b><i>rise over run<\/i><\/b><span style=\"font-weight: 400;\">.\u00a0\u00a0<\/span><\/p>\r\n<p><b>Facilitator Note<\/b><span style=\"font-weight: 400;\">: After each prompt, give students an opportunity to turn and talk with a partner before asking a few students to share their thoughts with the whole-group. <\/span><\/p>\r\n<h2><b>Visual Number Talk Prompt<\/b><\/h2>\r\n<p><span style=\"font-weight: 400;\">Student prompt:<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>What do you know about this pattern?<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>Jot down everything you know.<\/i><\/b><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2178 size-large\" src=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-1024x576.jpeg\" alt=\"Linear Equations From A Graph With Rise Over Run\" width=\"1024\" height=\"576\" srcset=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-1024x576.jpeg 1024w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-300x169.jpeg 300w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-768x432.jpeg 768w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-1536x864.jpeg 1536w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-1104x621.jpeg 1104w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-366x205.jpeg 366w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-534x300.jpeg 534w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-165x92.jpeg 165w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-249x140.jpeg 249w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-232x130.jpeg 232w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-344x193.jpeg 344w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-533x300.jpeg 533w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-405x228.jpeg 405w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001-139x78.jpeg 139w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-02-Math-Is-Visual-Prompt-image-001.jpeg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\r\n<p><span style=\"font-weight: 400;\">Give students an opportunity to turn and talk as well as share out to the group before presenting them with these questions:<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>What type of pattern is this, how do you know?<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>What context could be represented by this pattern?<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>What is the pattern rule?<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>Predict another value in this pattern not represented by this graph. Defend your answer. <\/i><\/b><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2179 size-large\" src=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-1024x576.jpeg\" alt=\"What type of pattern is this and what is the pattern rule?\" width=\"1024\" height=\"576\" srcset=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-1024x576.jpeg 1024w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-300x169.jpeg 300w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-768x432.jpeg 768w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-1536x864.jpeg 1536w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-1104x621.jpeg 1104w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-366x205.jpeg 366w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-534x300.jpeg 534w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-165x92.jpeg 165w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-249x140.jpeg 249w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-232x130.jpeg 232w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-344x193.jpeg 344w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-533x300.jpeg 533w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-405x228.jpeg 405w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002-139x78.jpeg 139w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-03-Math-Is-Visual-Prompt-image-002.jpeg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\r\n<p><span style=\"font-weight: 400;\">Students may have already shared that this graph was a decreasing linear relationship previously, however it might be helpful to ensure that all students recognize this and understand <\/span><b><i>why <\/i><\/b><span style=\"font-weight: 400;\">that is true. <\/span><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2180 size-large\" src=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-1024x576.jpeg\" alt=\"Constructing linear equations with a graph of a decreasing linear relationship\" width=\"1024\" height=\"576\" srcset=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-1024x576.jpeg 1024w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-300x169.jpeg 300w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-768x432.jpeg 768w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-1536x864.jpeg 1536w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-1104x621.jpeg 1104w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-366x205.jpeg 366w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-534x300.jpeg 534w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-165x92.jpeg 165w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-249x140.jpeg 249w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-232x130.jpeg 232w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-344x193.jpeg 344w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-533x300.jpeg 533w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-405x228.jpeg 405w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003-139x78.jpeg 139w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-04-Math-Is-Visual-Prompt-image-003.jpeg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\r\n<p><span style=\"font-weight: 400;\">Students may have also shared different contexts that could be represented by this relationship. Be intentional about ensuring students know that many contexts could work for this relationship and the one we will share here is not the <\/span><b><i>right <\/i><\/b><span style=\"font-weight: 400;\">context, but rather just one of many possibilities that could work.\u00a0<\/span><\/p>\r\n<p><span style=\"font-weight: 400;\">When students share the pattern rule, be mindful to listen for students referencing characteristics of this particular decreasing linear relationship. Are students referencing the initial value or rate of change without necessarily realizing it? Be sure to highlight or help them emerge these ideas based on what they share from the graph.\u00a0\u00a0<\/span><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2181 size-large\" src=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-1024x576.jpeg\" alt=\"Decreasing linear relationship graph using rise over run\" width=\"1024\" height=\"576\" srcset=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-1024x576.jpeg 1024w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-300x169.jpeg 300w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-768x432.jpeg 768w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-1536x864.jpeg 1536w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-1104x621.jpeg 1104w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-366x205.jpeg 366w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-534x300.jpeg 534w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-165x92.jpeg 165w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-249x140.jpeg 249w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-232x130.jpeg 232w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-344x193.jpeg 344w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-533x300.jpeg 533w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-405x228.jpeg 405w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004-139x78.jpeg 139w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-05-Math-Is-Visual-Prompt-image-004.jpeg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\r\n<p><span style=\"font-weight: 400;\">Although some students may use the language of something like:<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>\u201cIt goes down by 3 every time.\u201d<\/i><\/b><\/p>\r\n<p><span style=\"font-weight: 400;\">Be sure to explicitly share how we might see this relationship in the graph. For example, noting that after 4 days, the number of acorns decreased by 20 acorns is an equivalent ratio to after 2 days, the number of acorns decreased by 10 acorns.\u00a0<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>-20 acorns <\/i><\/b><span style=\"font-weight: 400;\">to <\/span><b><i>4 days<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>-10 acorns <\/i><\/b><span style=\"font-weight: 400;\">to <\/span><b><i>2 days<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>-5 acorns <\/i><\/b><span style=\"font-weight: 400;\">to <\/span><b><i>1 day<\/i><\/b><\/p>\r\n<p><span style=\"font-weight: 400;\">These equivalent <\/span><b><i>ratios <\/i><\/b><span style=\"font-weight: 400;\">can be used to reveal a <\/span><b><i>rate<\/i><\/b><span style=\"font-weight: 400;\"> through <\/span><b><i>partitive division<\/i><\/b><span style=\"font-weight: 400;\">. In this case, the rate would be <\/span><b><i>-5 acorns per day<\/i><\/b><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\r\n<p><span style=\"font-weight: 400;\">Often times, when using the <\/span><b><i>equivalent ratios <\/i><\/b><span style=\"font-weight: 400;\">to help model the <\/span><b><i>rate of change <\/i><\/b><span style=\"font-weight: 400;\">on a graph, mathematicians often call this strategy \u201c<\/span><b><i>rise over run<\/i><\/b><span style=\"font-weight: 400;\">\u201d since you are using the change in the dependent variable (y) partitively divided by the change in the independent variable (x).<\/span><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2184 size-large\" src=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-1024x576.jpeg\" alt=\"Linear Equation of decreasing linear graph\" width=\"1024\" height=\"576\" srcset=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-1024x576.jpeg 1024w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-300x169.jpeg 300w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-768x432.jpeg 768w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-1536x864.jpeg 1536w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-1104x621.jpeg 1104w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-366x205.jpeg 366w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-534x300.jpeg 534w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-165x92.jpeg 165w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-249x140.jpeg 249w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-232x130.jpeg 232w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-344x193.jpeg 344w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-533x300.jpeg 533w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-405x228.jpeg 405w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005-139x78.jpeg 139w, https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-06-Math-Is-Visual-Prompt-image-005.jpeg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\r\n<p><span style=\"font-weight: 400;\">By extending the pattern backwards to day 0, we can reveal the <\/span><b><i>initial value <\/i><\/b><span style=\"font-weight: 400;\">(y-intercept) and highlight a pattern rule of:<\/span><\/p>\r\n<p><b><i>Starts with 60 acorns, removing 5 acorns each day<\/i><\/b><\/p>\r\n<p><span style=\"font-weight: 400;\">Or, as algebraic expressions such as:<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>60 &#8211; 5d<\/i><\/b><span style=\"font-weight: 400;\">\u00a0\u00a0<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>-5d + 60<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>-5x + 60<\/i><\/b><\/p>\r\n<p><span style=\"font-weight: 400;\">Or, algebraic equations such as:<\/span><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>N = 60 &#8211; 5d<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>N = -5d + 60<\/i><\/b><\/p>\r\n<p style=\"padding-left: 40px;\"><b><i>y = -5x + 60<\/i><\/b><\/p>\r\n<h2>Want to Explore These Concepts &amp; Skills Further?<\/h2>\r\n<p><b><i>Two (2) additional number talk prompts are available<\/i><\/b><span style=\"font-weight: 400;\"> in <\/span><a href=\"https:\/\/learn.makemathmoments.com\/task\/acorn-hoarding-day4\"><span style=\"font-weight: 400;\">Day 4<\/span><\/a><span style=\"font-weight: 400;\"> of the <\/span><a href=\"https:\/\/learn.makemathmoments.com\/task\/acorn-hoarding\"><span style=\"font-weight: 400;\">Acorn Hoarding problem based math unit<\/span><\/a><span style=\"font-weight: 400;\"> you can dive into now.\u00a0<\/span><\/p>\r\n<p><span style=\"font-weight: 400;\">Why not start from the beginning of this contextual 5-day unit of <\/span><a href=\"https:\/\/learn.makemathmoments.com\/tasks\"><span style=\"font-weight: 400;\">real world lessons<\/span><\/a><span style=\"font-weight: 400;\"> from the Make Math Moments Problem Based Units page.<\/span><\/p>\r\n<p><span style=\"font-weight: 400;\">Did you use this in your classroom or at home? How\u2019d it go? Post in the comments!<\/span><\/p>\r\n<p><b>Math IS Visual. Let\u2019s teach it that way.<\/b><\/p>","protected":false},"excerpt":{"rendered":"<p>Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.<\/p>\n","protected":false},"author":1,"featured_media":2174,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[68,23],"tags":[50,61,51,52,53,32,33,34],"series":[54,94,95],"class_list":["post-2173","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra","category-number","tag-grade-2","tag-grade-3","tag-grade-4","tag-grade-5","tag-grade-6","tag-grade-7","tag-grade-8","tag-grade-9","series-number-talk","series-patterning","series-proportional-relationships"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Constructing Linear Equations From A Graph With Rise Over Run<\/title>\n<meta name=\"description\" content=\"Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Constructing Linear Equations From A Graph With Rise Over Run\" \/>\n<meta property=\"og:description\" content=\"Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\" \/>\n<meta property=\"og:site_name\" content=\"Math Is Visual\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/mathisvisual\" \/>\n<meta property=\"article:published_time\" content=\"2022-05-31T12:00:07+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-11-22T16:17:16+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif\" \/>\n\t<meta property=\"og:image:width\" content=\"640\" \/>\n\t<meta property=\"og:image:height\" content=\"400\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/gif\" \/>\n<meta name=\"author\" content=\"kylepearce3\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@mathisvisual\" \/>\n<meta name=\"twitter:site\" content=\"@mathisvisual\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"kylepearce3\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\"},\"author\":{\"name\":\"kylepearce3\",\"@id\":\"https:\/\/mathisvisual.com\/#\/schema\/person\/b47a49936e6a402ef530c1d283550a6e\"},\"headline\":\"Constructing Linear Equations From A Graph With Rise Over Run\",\"datePublished\":\"2022-05-31T12:00:07+00:00\",\"dateModified\":\"2022-11-22T16:17:16+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\"},\"wordCount\":856,\"commentCount\":0,\"image\":{\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif\",\"keywords\":[\"Grade 2\",\"Grade 3\",\"Grade 4\",\"Grade 5\",\"Grade 6\",\"Grade 7\",\"Grade 8\",\"Grade 9\"],\"articleSection\":[\"Algebra\",\"Number Sense\"],\"inLanguage\":\"en-CA\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\",\"url\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\",\"name\":\"Constructing Linear Equations From A Graph With Rise Over Run\",\"isPartOf\":{\"@id\":\"https:\/\/mathisvisual.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif\",\"datePublished\":\"2022-05-31T12:00:07+00:00\",\"dateModified\":\"2022-11-22T16:17:16+00:00\",\"author\":{\"@id\":\"https:\/\/mathisvisual.com\/#\/schema\/person\/b47a49936e6a402ef530c1d283550a6e\"},\"description\":\"Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#breadcrumb\"},\"inLanguage\":\"en-CA\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-CA\",\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage\",\"url\":\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif\",\"contentUrl\":\"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif\",\"width\":640,\"height\":400,\"caption\":\"Decreasing linear relationship graph using rise over run\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathisvisual.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Constructing Linear Equations From A Graph With Rise Over Run\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathisvisual.com\/#website\",\"url\":\"https:\/\/mathisvisual.com\/\",\"name\":\"Math Is Visual\",\"description\":\"Let&#039;s teach it that way.\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathisvisual.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-CA\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathisvisual.com\/#\/schema\/person\/b47a49936e6a402ef530c1d283550a6e\",\"name\":\"kylepearce3\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-CA\",\"@id\":\"https:\/\/mathisvisual.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/793650925788278dc82a6c7461d8fa6fff2e7680c114f04295ad0479a8ca15c9?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/793650925788278dc82a6c7461d8fa6fff2e7680c114f04295ad0479a8ca15c9?s=96&d=mm&r=g\",\"caption\":\"kylepearce3\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Constructing Linear Equations From A Graph With Rise Over Run","description":"Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/","og_locale":"en_US","og_type":"article","og_title":"Constructing Linear Equations From A Graph With Rise Over Run","og_description":"Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.","og_url":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/","og_site_name":"Math Is Visual","article_publisher":"https:\/\/www.facebook.com\/mathisvisual","article_published_time":"2022-05-31T12:00:07+00:00","article_modified_time":"2022-11-22T16:17:16+00:00","og_image":[{"width":640,"height":400,"url":"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif","type":"image\/gif"}],"author":"kylepearce3","twitter_card":"summary_large_image","twitter_creator":"@mathisvisual","twitter_site":"@mathisvisual","twitter_misc":{"Written by":"kylepearce3","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#article","isPartOf":{"@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/"},"author":{"name":"kylepearce3","@id":"https:\/\/mathisvisual.com\/#\/schema\/person\/b47a49936e6a402ef530c1d283550a6e"},"headline":"Constructing Linear Equations From A Graph With Rise Over Run","datePublished":"2022-05-31T12:00:07+00:00","dateModified":"2022-11-22T16:17:16+00:00","mainEntityOfPage":{"@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/"},"wordCount":856,"commentCount":0,"image":{"@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage"},"thumbnailUrl":"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif","keywords":["Grade 2","Grade 3","Grade 4","Grade 5","Grade 6","Grade 7","Grade 8","Grade 9"],"articleSection":["Algebra","Number Sense"],"inLanguage":"en-CA","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/","url":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/","name":"Constructing Linear Equations From A Graph With Rise Over Run","isPartOf":{"@id":"https:\/\/mathisvisual.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage"},"image":{"@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage"},"thumbnailUrl":"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif","datePublished":"2022-05-31T12:00:07+00:00","dateModified":"2022-11-22T16:17:16+00:00","author":{"@id":"https:\/\/mathisvisual.com\/#\/schema\/person\/b47a49936e6a402ef530c1d283550a6e"},"description":"Visual Number Talk Prompts emerging Linear Equations From A Graph using Rise Over Run to make predictions and solve equations.","breadcrumb":{"@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#breadcrumb"},"inLanguage":"en-CA","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathisvisual.com\/linear-equations-with-graphs\/"]}]},{"@type":"ImageObject","inLanguage":"en-CA","@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#primaryimage","url":"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif","contentUrl":"https:\/\/mathisvisual.com\/wp-content\/uploads\/2022\/05\/Acorn-Hoarding-Day-4-Purposeful-Practice-07-Math-Is-Visual-Prompt-GIF.gif","width":640,"height":400,"caption":"Decreasing linear relationship graph using rise over run"},{"@type":"BreadcrumbList","@id":"https:\/\/mathisvisual.com\/linear-equations-with-graphs\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathisvisual.com\/"},{"@type":"ListItem","position":2,"name":"Constructing Linear Equations From A Graph With Rise Over Run"}]},{"@type":"WebSite","@id":"https:\/\/mathisvisual.com\/#website","url":"https:\/\/mathisvisual.com\/","name":"Math Is Visual","description":"Let&#039;s teach it that way.","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathisvisual.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-CA"},{"@type":"Person","@id":"https:\/\/mathisvisual.com\/#\/schema\/person\/b47a49936e6a402ef530c1d283550a6e","name":"kylepearce3","image":{"@type":"ImageObject","inLanguage":"en-CA","@id":"https:\/\/mathisvisual.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/793650925788278dc82a6c7461d8fa6fff2e7680c114f04295ad0479a8ca15c9?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/793650925788278dc82a6c7461d8fa6fff2e7680c114f04295ad0479a8ca15c9?s=96&d=mm&r=g","caption":"kylepearce3"}}]}},"_links":{"self":[{"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/posts\/2173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/comments?post=2173"}],"version-history":[{"count":4,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/posts\/2173\/revisions"}],"predecessor-version":[{"id":2185,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/posts\/2173\/revisions\/2185"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/media\/2174"}],"wp:attachment":[{"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/media?parent=2173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/categories?post=2173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/tags?post=2173"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/mathisvisual.com\/wp-json\/wp\/v2\/series?post=2173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}