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Welcome back to our continuing exploration of how to bring real conceptual reasoning to questions students might encounter on a high school equivalency test. It has been a chilly winter where I am in Massachusetts, so let\u2019s look at a wintry problem this month. (Maybe by the time you\u2019re reading this, you\u2019ll be thinking more about flowers than icicles!)<\/p>\n\n\n\n
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<\/figure>Pause to reflect. How many strategies can you think of? What visuals could you use to help solve this? Also ask yourself, what skills and understandings do students really need<\/em> to be able to answer this?<\/strong><\/p>\n<\/div><\/div>\n\n\n\n As you were pondering this question, you may have wondered what domain this comes from. Is this a question about fraction operations? Is it a question about proportional reasoning? As with most substantive problems, this doesn\u2019t boil down to a single application of a single skill. A student facing this problem will need to draw on skills from different math lessons. This is an important thing to consider as we plan our lessons \u2014 if we integrate topics, concepts, and skills, students will know how to do the same on a test. If we teach and assess a single skill at a time, we are not preparing students to use their math for problem-solving on a test or in life.<\/p>\n\n\n\n 1. Estimate.<\/strong> It\u2019s not easy to make sense of a rate like 2\/5 of an inch per minute, but a student who has a strong sense of benchmark fractions may reason that 2\/5 is a little less than \u00bd (because 2 is a little less than half of 5). Knowing that the icicle is growing at a rate of a little less than half of an inch each minute, a student may reason that it grows a little less than 1 inch every 2 minutes. 2\u00bd hours is 150 minutes, so the icicle would grow a little less than 1 inch 75 times (because 150 minutes is 75 groups of 2 minutes). <\/p>\n\n\n\n 2. Take small steps.<\/strong> Knowing the rate of growth per minute means that we know how much the icicle grows every<\/em> minute. A student might make a chart to keep track of the growth of the icicle minute by minute:<\/p>\n\n\n\n A few notes about this approach:<\/p>\n\n\n\n 3. Chunk up the time.<\/strong> A student who arrives at the conclusion that the icicle grows 2 inches in 5 minutes may also start to think about how many groups of 5 minutes are in 2\u00bd hours:<\/p>\n\n\n\n There are 12 groups of 5 minutes in 1 hour, so there are 24 groups of 5 minutes in 2 hours<\/u>.<\/em><\/p> There are 12 groups of 5 minutes in 1 hour, so there are 6 groups of 5 minutes in half an hour.<\/u><\/em><\/p> Therefore, there are 24 + 6 groups of 5 minutes in 2\u00bd<\/em> hours.<\/em><\/p> In each of those 30 groups of 5 minutes, the icicle grows 2 inches, so it grows 2 inches 30 times.<\/em><\/p><\/blockquote>\n<\/div>\n<\/div>\n\n\n\n 4. Multiply the number of minutes by the amount the icicle grows in each minute.<\/strong> Of course we can use a procedure to multiply 2\/5 of an inch per minute by 150 minutes, but let\u2019s look at what it can look like with a picture instead. We\u2019ll use bar models to make 150 groups of 2\/5 (or at least to get an idea of what it might look like if we made 150 groups.)<\/p>\n\n\n\n How would you continue this multiplication model? What connections can you make between this visual and the chunking strategies above? What connections can you make to a procedure you know for multiplying fractions?<\/p>\n\n\n\n Several of these approaches involve reasoning proportionally in increments to get to a final answer. One thing I love about this kind of approach is that it gives the reasoner flexibility to choose the size of the steps they want to take. Look back at strategies 2 and 3. Would you make the same decisions about when to jump to bigger chunks or what size chunks to use? Why or why not? What seems easiest or most efficient to you may seem harder or less efficient to someone else. When I reason this way, I don\u2019t always take the same size steps \u2014 it depends on the numbers I\u2019m dealing with and on my whims! Empowering students to be flexible in their proportional reasoning makes them stronger reasoners and positions them as thinkers instead of mere followers of someone else\u2019s rules.<\/p>\n\n\n\n Sarah<\/em> Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. Sarah\u2019s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center<\/a> at 91±¬ΑΟ<\/a> includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network<\/a>.<\/em><\/p>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":" by Sarah Lonberg-Lew<\/p>\n Welcome to the latest installment of our monthly series, \u201cWill This Be on the Test?\u201d Each month, we\u2019ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.<\/em><\/p>\n![\"\"](\"https:\/\/www.terc.edu\/adultnumeracycenter\/wp-content\/uploads\/sites\/28\/2022\/02\/WTBotT_Img2_Mar2022.png\")
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