.... ah, so by extension, for say a fraction equivalent to 3/7, they might prefer to choose a random numerator, × it by 7 to get the denominator, then go back and × the numerator by 3.
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I hadn’t thought that far ahead. What you describe is very much ratio-based thinking. Starting with x, you’re creating the fraction 3x/7x. I think you could get a really nice lesson out of that which would help develop a ratio-based (rational!) conceptual understanding
How about this: We can multiply both fractions by 11, which wouldn't change the size relationship. This gives 55/7 vs 8. We know that 55/7 is less than 8 bc it is 7-point-something. Hence the original 5/7 is less than 8/11.
True and I like this! But I’m trying to work within a fractions-as-ratios mindset … so I’m trying to do it by thinking about the ratio of the numerator to the denominator, rather than thinking about the fractions as single (part-whole) entities. I’m trying to think like @germinalmaths.bsky.social!
Yes, and you might be right. But I'm struck by the difficulty of finding "5/7 of 11". Even with my facility with numbers, I'm not sure what this works out to be. But 55/7 - ah, that's smaller than 8.
I'm going off my last layout ... feels like I'm avoiding working 'within' the fractions, avoiding dealing with the ratio of numerator to denominator. Perhaps this is a better graphic to illustrate your thought processes? (Top: before scaling denominator. Bottom: after scaling denominator by 5/7).
The mapping diagram is an interesting representation.
As R is below G, the scale factor that maps Num onto Den is larger for 5/7 than for 8/11, so 5/7 < 8/11
The gradient of each line corresponds to the reciprocal of each fraction. This illustrates another numerical strategy that is sometimes useful when comparing two fractions a and b. If 1/a > 1/b, then a < b and vice versa. Not really helpful in this case (the picture is, though).
This … rescales the fraction 5/7 to 1 unit … and similarly scales 8/11 to … 8/(55/7) … and then by comparison of numerator and denominator this is either greater or less than 1?
Hmm. I like it … though I might lay it out like this.
This is not quite my thinking. What you have done, though is essentially to divide 8/11 by 5/7.
This is another general strategy for comparing two fractions a and b: if a/b > 1 then a > b and vice versa.
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As R is below G, the scale factor that maps Num onto Den is larger for 5/7 than for 8/11, so 5/7 < 8/11
This … rescales the fraction 5/7 to 1 unit … and similarly scales 8/11 to … 8/(55/7) … and then by comparison of numerator and denominator this is either greater or less than 1?
Hmm. I like it … though I might lay it out like this.
This is another general strategy for comparing two fractions a and b: if a/b > 1 then a > b and vice versa.