tag:blogger.com,1999:blog-6651514617266100245.post5861183154616681215..comments2024-01-03T04:58:04.221-05:00Comments on I Hope This Old Train Breaks Down...: A different approach to linear equations?Unknownnoreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6651514617266100245.post-58597460222695488052011-09-07T00:43:51.573-04:002011-09-07T00:43:51.573-04:00I love this approach. I agree with brainopennow – ...I love this approach. I agree with brainopennow – the equations now make sense (and they will make sense in different ways as students visualize the figures differently). Not at all rote learning. Without knowing your grade 7 curriculum, I, too, would follow this up with graphing. As an extension, students could try to find the equation of quadratic patterns. Some interesting connections to be made here. See http://wp.me/p1OFpp-3g for more of my thoughts on this approach. Thanks for sharing!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-35221019620149347212011-09-06T13:48:40.651-04:002011-09-06T13:48:40.651-04:00@Bowen Thanks for the feedback! I've already d...@Bowen Thanks for the feedback! I've already done the introductory lessons with the dots with Grade 7, so the 1 + 3 + 3 + 3 ... revisions may have to wait until next year. But, I do like the recommendation of including Stage 0 in the future. I'll do that when we introduce graphing, so they can see the graphical/equation/visual connections.<br /><br />David Cox also uses something similar, but he uses fractions of a circle as the changing rates. That's also a good next step, once the kids get comfortable graphing/writing equations/making tables for integer coeffs.Mimihttp://untilnextstop.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-78116046643259299832011-09-05T16:18:21.260-04:002011-09-05T16:18:21.260-04:00I like it! Reminds me of the genre of problems se...I like it! Reminds me of the genre of problems seemingly called "pile patterns", of which you can find many examples thru Google search. Typically the piles are made of little contiguous squares, which leads to an easier (probably) extension to using area formulas. But it's good to see the counting in both ways.<br /><br />This also reminds me that I need to share my "pile patterns" stuff since my kids (yes, 11th/12th grade) really latched onto this and found that the equations "made sense".Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-67927657100693882412011-09-05T16:01:08.347-04:002011-09-05T16:01:08.347-04:00At the stage where they are still gathering, it ca...At the stage where they are still gathering, it can be a good idea to write it out more literally: 1 + 3 + 3 + 3, then 1 + 3 + 3 + 3 + 3, then... Eventually a student will point out pulling the 3s together. You can later use the same concept when combining like terms on 1 + x + x + x.<br /><br />I'd also encourage starting with a "Stage 0", even if it seems clunky at first, since it links nicely to graphs and equation forms. For example, the "16" that magically appears in your #9 wouldn't be magical if you had a Stage 0, and the rule in #8 (12 + 4n) makes a lot more sense if there is a 12 in the table.<br /><br />(This same issue comes up in sequences and series: sequences that start with "term 1" end up getting messy rules with (n-1) in them, while sequences that start with "term 0" don't. But not in 7th grade!)<br /><br />Good luck and good work! Seems like the next step is to graph "dots" against "stage". Make sure that when kids write y = 16-2x (or whatever) they can clearly say what x and y stand for. Also bring in some non-examples, like square patterns, to make it clear that not all patterns are linear (a trap kids fall into quickly).Anonymousnoreply@blogger.com