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Partitioning fractionsPartitioning fractions means being able to divide a group or set of things into equal parts. I underline the idea of equal parts. And when I say equal I mean equal! If there is one overall rule in mathematics is that there is a zero tolerance for mistakes! A tiny imperfection or uncorrectness and you are in for troubles...so make those parts equals! Let's go back to partitioning. Dividing into equal parts, we sere saying.... That is the first thing to learn. Afterwards, comes the difficult task to actualy name those many "equal parts" as a fraction, and precisely according to the number of those parts! A quantum leap for any student! So partitioning is this: Dividing into equal parts and naming those parts as a fraction!
If you fail in partitioning you will fail in establishing equivalent fractions and end up saying that 7/8 = 5/6 because "they look the same".[1]
In the above picture, the fractions (7/8 on the left and 5/6 on the righthand side) "look" identical but are not! Partitioning was wrongly done! In the picture below, the same picture but enlarged (150% bigger, that is one and a half times more than the original image, hey that's a ratio!) to show the unprecisions in drawing. Each color represents a different width of the bar, even if it is a minor difference, it matter a lot!
505. Indeed, the aim of teaching [mathematics] should be MAGNUS, PHILIP. source: Memorabilia mathematica; or, The philomath's quotationbook  Moritz, Robert Édouard, 18681940
Source: 1. Teaching Fractions And Ratios For Understanding: Essential Content Knowledge And Instructional...

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