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Welcome back to our continuing exploration of how to bring real conceptual reasoning to questions students might encounter on a standardized test. <\/p>\n\n\n\n
I have argued throughout this series that many standardized test math questions can be approached visually and conceptually and that students who may not have studied the specific content the question is targeting may still have a solution path if they are disposed to make sense and think flexibly. In general, the more concrete and contextualized a question is, the more likely it is that multiple creative and flexible solution paths are available. I encourage students to see story problems as a gift because the context opens up more avenues to sense-making. However, questions that are more abstract or appear to involve higher level concepts are also sometimes<\/em> accessible through a sense-making approach even when students have not learned the relevant procedures.<\/p>\n\n\n\n Here’s one such question for this month:<\/a><\/p>\n\n\n How can you approach this question in a way that makes sense to you<\/em>? What conceptual understandings or visual tools can you bring to bear? What mathematical concepts do students really<\/em> need to be able to tackle this problem? <\/strong>How might your real-world experience help you reason about this?<\/strong><\/p>\n<\/div><\/div>\n\n\n\n If you look in an algebra textbook or test prep book or website, it is likely to tell you that the<\/em> way to solve an equation like this is by using inverse operations and isolating x<\/strong><\/em>. That works if it is an approach you understand and are comfortable with. But what if you haven\u2019t learned all those algebra rules or they get muddled during a math panic and can\u2019t remember what you are supposed to do?<\/p>\n\n\n\n This question is<\/em> abstract and not accessible to every learner, but the understandings needed to solve it may be fewer than you think. My challenge to you is to solve this relying only on these basic understandings:<\/p>\n\n\n\n How can you make sense of the equation using these understandings and find a value for x<\/em><\/strong> that makes it true?<\/p>\n\n\n\n Here are some approaches:<\/p>\n\n\n\n 1. Make sense of the story. Chunk it up. <\/strong>Equations and mathematical notation in general are shorthand. They are ways of recording relationships or processes in symbols instead of words. But the same stories can be told in words. Here\u2019s a way of telling the story of this equation:<\/p>\n\n\n\n There\u2019s some number that if I multiply it by 6 and then subtract 3 from the answer, I\u2019ll get 21.<\/strong><\/em><\/p>\n\n\n\n A useful follow-up question to this reframing is:<\/p>\n\n\n\n What number gives me 21 if I subtract 3 from it?<\/strong><\/em><\/p>\n\n\n\n One way to answer that question is to make use of the inverse relationship of addition and subtraction and add 3 to 21, but there are other ways, too. A student might picture a number line (more on that later) or pick a number that\u2019s a little bigger than 21 and try counting down by 3 to see if they get to 21, or they might think, \u201cI know 4 \u2013 3 is 1, so 24 \u2013 3 is 21.\u201d <\/p>\n\n\n\n Having answered that question, a student can then simplify the original story like this:<\/p>\n\n\n\n There\u2019s some number that, if I multiply it by 6, I will get 24.<\/strong><\/em><\/p>\n\n\n\n You might be thinking that this approach is not that different from a traditional approach of using inverse operations to isolate x<\/em><\/strong>. The difference is in its being more concrete. Using words and the idea of \u201csome number\u201d is more accessible to students whose algebraic thinking has not yet reached the level of abstraction that allows them to read and make sense of symbols as easily as they do words.<\/p>\n\n\n\n 2. Use a story table. <\/strong>Another approach that makes the story of the equation visible is a story table that separates out the action into one step per column. A story table for this equation could look like this:<\/p>\n\n\n\n![\"What](\"https:\/\/www.terc.edu\/adultnumeracycenter\/wp-content\/uploads\/sites\/28\/2025\/12\/WTBotT_Dec2025_Img1.png\")
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![\"A](\"https:\/\/www.terc.edu\/adultnumeracycenter\/wp-content\/uploads\/sites\/28\/2025\/12\/Screenshot-2025-12-05-at-10.51.57-AM.png\")
<\/figure>\n<\/div>\n\n\n![\"A](\"https:\/\/www.terc.edu\/adultnumeracycenter\/wp-content\/uploads\/sites\/28\/2025\/12\/Screenshot-2025-12-05-at-10.52.16-AM.png\")
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