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I have some comments about the RP Progressions that I would like to put out there. I welcome any and all input. Sue
When I looked at “Progression on Ratios and Proportional Relationships” I noticed no mention of the importance of ordering a ratio expression when it is placed into a table. My thought here is that the order in which two quantities are related makes a difference when these two quantities are displayed in a table and then later graphed. Further, the language surrounding the words that describe a ratio should be consistent so that students can more easily discern this relationship. When I look at standards 6.RP. 1 – 3, and standard 6.EE.9, I note a disconnect. I found this same disconnect between pages 5 and 6 of the RP Progressions article and also within a recent Texas Instruments webinar. On page 6 of the RP Progressions article and in the TI webinar, I noticed that the ratio expression was not properly treated when it was placed into a table where units were attached.
The first treatment of the ratio expression on page 5 of the RP Progressions article differs from the second treatment of the ratio expression on page 6. I believe that the first treatment of the ratio: ““for every 5 cups grape juice, mix in 2 cups peach juice” was correctly represented within the table with grape juice being shown within the first column and then later graphed as the independent variable on the x. However, I noticed that this was later reversed in the tables shown on page 6 when the ratios: “1 cup red paint for every 3 cups yellow paint and … 3 cups red for every 5 cups yellow” were arranged within the table with red paint as the independent variable.
In a like manner I found similar flip-flopped reversals of ratios displayed within a graph during a recently viewed Texas Instruments on-demand webinar entitled: “Deciphering Ratios with TI- Inspire Technology: Are They Fractions?” Fifty minutes into this webinar, when it came time to display the rate 3m for every 2 seconds in a graph, the points were labelled in reverse order with distance listed first. I have captured this in a screen shot attachment below. I am wondering if this was done to maintain the ratio as it was originally read. I am also wondering if this is an ideal representation.
I am also wondering if we shouldn’t explicitly teach rate as a special type of ratio in which units are attached and order matters. If we discuss this order, the language clues, semantics, and relationship contexts prior to placing a ratio into a table and graphing it, the potential for later student confusion might be avoided.
Any thoughts on this???
I would argue that the ratio and proportions standards when viewed in light of the overarching mathematical practices and underlying grade 5 standards 5.OA.3 and 5.G.2 would support an earlier exposure to “negative slope” relationships than grade 8. If we support the traditionally less fluent operations of subtraction and division through grade 5 exposure to descending patterns within tables, student facility with these operations can not only be remediated, but a foundation for the understanding of negative proportional relationships can be laid. In grade 6, this can then be further supported by providing situational contexts for negative slope that the students can relate to such as tracking a runner’s distance from Home Base in the problem “Running Home From Third Base”. Grade 6 student understanding of negative slope has been shown to be easily facilitated by combining this earlier introduction to descending table patterns with a subsequent physical modeling of decreasing distance over time. When this physical modeling was then combined with freeze-framed second-by-second representations of this motion on a number line students had little difficulty conceptualizing and interacting with the tabular and graphical representations of this negative slope scenario.
