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Now, I can visualize how $a/b \times a/d=ab/cd$ for partial fractions, but I can't for the life of me visualize how they can work for multiplying an improper fraction anywhere (i.e. either multiplying a partial fraction with an improper one or two improper fractions).

Let's say, $1/5\times 2/3. $ Let's visualize it to get a good idea of what I'm talking about. enter image description here Now, let's "overlay" the denominator 3, by cutting that 5 up into 3 parts, making it $5\times3. $ enter image description here And now, let's get 2/3 of 1/5. enter image description here

Now, as you can see, the shaded in "overlaid" part is $2/3 \times 1/5,$ or $2/15.$ However, I want to know how this could apply to both multiplying partial fractions with improper fractions (e.g. $1/3\times 4/3$) and also two improper fractions. I feel like both have a different way of how they work. Thank you.

LinguisticChiyo
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  • Converting improper fractions into mixed fractions might help. e.g. $\frac{4}{3}=1\frac{1}{3}$ – Aman Kushwaha Aug 28 '21 at 03:53
  • Just add extra white spaces. Each block of white should be the same size. Then cut and shade them all in the same way, for each fraction. They will be separated, but close enough to get the idea. – Bafs Aug 28 '21 at 07:03
  • Related, perhaps helpful: https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127776#1127776 – Ethan Bolker Aug 29 '21 at 13:59

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The thick green segments indicate the unit lengths in the indicated directions. The figures on the right indicate the "value" of the indicated square or rectangle.

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Steven Alexis Gregory
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