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 - ThreadSky

germinalmaths.bsky.social • 28 days ago

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

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germinalmaths.bsky.social•28 days ago

Regarding your written example, I would find 5/7 of 11 and compare it with 8. Alternatively, find 8/11 of 7 and compare with 5.

tcorica.bsky.social•28 days ago

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.

segarrogers.bsky.social•28 days ago

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!

tcorica.bsky.social•28 days ago

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.

germinalmaths.bsky.social•28 days ago

But 5/7 of 11 is 55/7.

tcorica.bsky.social•28 days ago

Yes! But I don't have a feel for whether 5/7 of 11 is going to be greater than 8 unless I do the (numerator) multiplication. I guess that's what I'm trying to say.

segarrogers.bsky.social•28 days ago

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).

profsmudge.bsky.social•28 days ago

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

germinalmaths.bsky.social•28 days ago

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).

segarrogers.bsky.social•28 days ago

I … think I understand?!

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.

germinalmaths.bsky.social•28 days ago

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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