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I decided to see if I could break down learning about the concept of initial value in linear equations based on the video from Lesson 36 using Stacks of Paper. Given 1 pack of paper = 4.95 cm tall and that 5 packs of paper on a table is 130.75 cm tall, find the height of the table.
U.I. The student adds 4.95 five times or multiplies 4.95 by 5 to get that 5 packs of paper is 24.75 cm tall. They then subtract 24.75 from 130.75 to find the height of the table is 106 cm.
C.I. The student makes a table of #packs vs. height from floor by starting with 5 pks plus table is 130.75, 4 pks is 125.8 and on down to 1 pk is 110.95 and 0 pk (just the table) is 106 cm. They think there must be a way to generalize this to any number of packs stacked on a table, but aren’t sure how to do this.
C.C. Student knows they can multiply #packs times hgt/pack and add the table height so they can write total height = #packs(4.95) + table height and then 130.75 = 5(4.95) + 106 which they can connect to y = mx + b once they are shown how.
U.C. Student automatically looks for dependent and independent variables and solves for initial value using y = mx + b whenever they are faced with a situation like this.
C.M When learning another function family, perhaps exponential functions, student postulates that the yintercept may be the initial value and tests this theory out.
Would appreciate feedback as I’m not sure if I’m mixing too much in and not making it granular enough.